OEIS/Eta products - Revision history

Gfis: /* Rogers-Ramanujan continued fraction r */ -> A7325

2025-05-31T09:19:21Z

Rogers-Ramanujan continued fraction r: -> A7325

← Older revision		Revision as of 09:19, 31 May 2025
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	EPSIG="[3,3;1,-1]" Q="-1/3" F=3		EPSIG="[3,3;1,-1]" Q="-1/3" F=3
	====Rogers-Ramanujan continued fraction r====		====Rogers-Ramanujan continued fraction r====
	[https://oeis.org/~~A003823 A003823~~] 1, 1, 0, -1, 0, 1, 1, -1, -2, 0, 2, 2, -1, -3, -~~1, 3~~		[https://oeis.org/A007325 A007325] 1, -1, 1, 0, -1, 1, -1, 1, 0, -1, 2, -3, 2, 0, -2, 4
	1+x/(1+x^2~~/(1+~~x^3~~/(1+x^4/(1+...)))~~)		Expansion of f(-x, -x^4) / f(-x^2, -x^3)

	=== Eisenstein series===		=== Eisenstein series===
	==== E_2 = Ramanujan P(q) ====		==== E_2 = Ramanujan P(q) ====

Gfis: Rogers-Ramanujan r

2025-05-31T08:08:07Z

Rogers-Ramanujan r

← Older revision		Revision as of 08:08, 31 May 2025
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	3 * (Euler transform of period 3: [-3,-3,-2])		3 * (Euler transform of period 3: [-3,-3,-2])
	EPSIG="[3,3;1,-1]" Q="-1/3" F=3		EPSIG="[3,3;1,-1]" Q="-1/3" F=3
			====Rogers-Ramanujan continued fraction r====
			[https://oeis.org/A003823 A003823] 1, 1, 0, -1, 0, 1, 1, -1, -2, 0, 2, 2, -1, -3, -1, 3
			1+x/(1+x^2/(1+x^3/(1+x^4/(1+...))))
	=== Eisenstein series===		=== Eisenstein series===
	==== E_2 = Ramanujan P(q) ====		==== E_2 = Ramanujan P(q) ====

Gfis: Product g.f.s

2024-10-17T18:43:31Z

Product g.f.s

← Older revision		Revision as of 18:43, 17 October 2024
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	[https://oeis.org/A000041 A000041] 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231		[https://oeis.org/A000041 A000041] 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231
	Euler transform of period 1: [1]		Euler transform of period 1: [1]
			G.f.: Product_{k>0} 1/(1-x^k)
	EPSIG="[1,-1]" Q="1/24"		EPSIG="[1,-1]" Q="1/24"
	====Dedekind eta η (without the q^(1/24) factor), Ramanujan f(-q)====		====Dedekind eta η (without the q^(1/24) factor), Ramanujan f(-q)====
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	a(0) = 1, for n >= 1: a(n) = 2 if n is a square, otherwise 0.		a(0) = 1, for n >= 1: a(n) = 2 if n is a square, otherwise 0.
	G.f.: eta(q^2)^5/(eta(q)*eta(q^4))^2		G.f.: eta(q^2)^5/(eta(q)*eta(q^4))^2
			G.f.: Product_{k>0} (1 - x^(2k))(1 + x^(2*k-1))^2.
	Euler transform of period 4: [2,-3,2,-1]		Euler transform of period 4: [2,-3,2,-1]
	EPSIG="[2,5;4,-2;1,-2]"		EPSIG="[2,5;4,-2;1,-2]"
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	a(n) = 1 if n is a triangular number, otherwise 0.		a(n) = 1 if n is a triangular number, otherwise 0.
	G.f.: q^(-1/8) * eta(q^2)^2/eta(q)		G.f.: q^(-1/8) * eta(q^2)^2/eta(q)
			G.f.: Product_{k>0} (1-x^(2k))/(1-x^(2k-1))
	Euler transform of period 2: [1,-1]		Euler transform of period 2: [1,-1]
	EPSIG="[2,2;1,-1]"		EPSIG="[2,2;1,-1]"
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	[https://oeis.org/A000700 A000700] 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5		[https://oeis.org/A000700 A000700] 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5
	G.f.: q^(1/24) * eta(q^2)^2/(eta(q)*eta(q^4))		G.f.: q^(1/24) * eta(q^2)^2/(eta(q)*eta(q^4))
			G.f.: Product_{k>=1} (1 + x^(2*k-1))
	Euler transform of period 4: [1,-1,1,0]		Euler transform of period 4: [1,-1,1,0]
	EPSIG="[2,2;4,-1;1,-1]"		EPSIG="[2,2;4,-1;1,-1]"
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	[https://oeis.org/A000594 A000594] 1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612		[https://oeis.org/A000594 A000594] 1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612
	G.f.: eta(q)^24		G.f.: eta(q)^24
			G.f.: x * Product_{k>=1} (1 - x^k)^24
	Euler transform of period 1: [-24]		Euler transform of period 1: [-24]
	EPSIG="[1,24]"		EPSIG="[1,24]"

Gfis: 1723168255201

2024-10-04T11:41:10Z

172316825520*1*

← Older revision		Revision as of 11:41, 4 October 2024
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	E_26: [https://oeis.org/A282356 A282356] a(0) = 657931, for n >= 1: a(n) = -24*sigma_25(n)		E_26: [https://oeis.org/A282356 A282356] a(0) = 657931, for n >= 1: a(n) = -24*sigma_25(n)
	E_28: [https://oeis.org/A282401 A282401] a(0) = 3392780147, for n >= 1: a(n) = 6960*sigma_27(n)		E_28: [https://oeis.org/A282401 A282401] a(0) = 3392780147, for n >= 1: a(n) = 6960*sigma_27(n)
	E_30: [https://oeis.org/A282182 A282182] a(0) = ~~172316825520~~, for n >= 1: a(n) = -171864*sigma_29(n)		E_30: [https://oeis.org/A282182 A282182] a(0) = 1723168255201, for n >= 1: a(n) = -171864*sigma_29(n)
	E_32: [https://oeis.org/A282540 A282540] a(0) = 7709321041217, for n >= 1: a(n) = 32640*sigma_31(n)		E_32: [https://oeis.org/A282540 A282540] a(0) = 7709321041217, for n >= 1: a(n) = 32640*sigma_31(n)
	The <code>a(0)</code> constants are given by <code>abs([https://oeis.org/A001067 A001067: Numerator of Bernoulli(2n)/(2n)])</code>, while the factors of <code>sigma_xx(n)</code> are found in <code>[https://oeis.org/A006863 A006863: Denominator of B_{2n}/(-4n</code>]		The <code>a(0)</code> constants are given by <code>abs([https://oeis.org/A001067 A001067: Numerator of Bernoulli(2n)/(2n)])</code>, while the factors of <code>sigma_xx(n)</code> are found in <code>[https://oeis.org/A006863 A006863: Denominator of B_{2n}/(-4n</code>]

Gfis: E_2k

2024-10-04T11:12:35Z

E_2k

← Older revision		Revision as of 11:12, 4 October 2024
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	G.f.: ((eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8)^2		G.f.: ((eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8)^2
	"[1,24;1,-8;2,-8]+256*[2,24;1,-8;2,-8]" squared		"[1,24;1,-8;2,-8]+256*[2,24;1,-8;2,-8]" squared
	==== ~~E_10~~, ~~E_12~~, ~~E_14~~, ~~etc~~. ====		==== E_2k, k=1..16 ====
			E_2: [https://oeis.org/A006352 A006352] a(0) = 1, for n >= 1: a(n) = -24*sigma_1(n)
			E_4: [https://oeis.org/A004009 A004009] a(0) = 1, for n >= 1: a(n) = 240*sigma_3(n)
			E_6: [https://oeis.org/A013973 A013973] a(0) = 1, for n >= 1: a(n) = -504*sigma_5(n)
			E_8: [https://oeis.org/A008410 A008410] a(0) = 1, for n >= 1: a(n) = 480*sigma_7(n)
	E_10: [https://oeis.org/A013974 A013974] a(0) = 1, for n >= 1: a(n) = -264*sigma_9(n)		E_10: [https://oeis.org/A013974 A013974] a(0) = 1, for n >= 1: a(n) = -264*sigma_9(n)
	E_12: [https://oeis.org/A029828 A029828] a(0) = 691, for n >= 1: a(n) = 65520*sigma_11(n)		E_12: [https://oeis.org/A029828 A029828] a(0) = 691, for n >= 1: a(n) = 65520*sigma_11(n)

Gfis: /* E_10, E_12, E_14, etc. */

2024-10-04T11:04:21Z

E_10, E_12, E_14, etc.

Gfis: E_10, E_12 etc.

2024-10-03T18:16:12Z

E_10, E_12 etc.

← Older revision		Revision as of 18:16, 3 October 2024
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	===Eta product signatures (EPSIG)===		===Eta product signatures (EPSIG)===
	The coefficient sequences for generating functions that are products of ~~the~~ eta function can be computed conveniently by an Euler transform of some periodic integer sequence.		The coefficient sequences for generating functions that are products of Dedkind's eta function can be computed conveniently by an Euler transform of some periodic integer sequence.
	Following Michael Somos, we describe an eta product by a matrix resp. a list of pairs ''(q<sub>i</sub>, e<sub>i</sub>)'' of the form		Following Michael Somos, we describe an '''eta product''' by a matrix resp. a list of pairs ''(q<sub>i</sub>, e<sub>i</sub>)'' of the form
	[q1,e1;q2,e2;q3,e3;...]		[q1,e1;q2,e2;q3,e3;...].
	The pairs are separated by ";".		The pairs are separated by ";".
	* The ''q<sub>i</sub>'' are the powers of the argument ''q'' inside the eta function , and		* The ''q<sub>i</sub>'' are the powers of the argument ''q'' inside the eta function , and
	* the ''e<sub>i</sub>'' are the powers of the eta functions. Only these can be negative.		* the ''e<sub>i</sub>'' are the powers of the eta functions. Only these can be negative.
	The following sections contain a number of examples.		We call such a list the ''signature'' ("EPSIG") of the eta product. The following sections contain a number of examples.

	The order of the pairs is irrelevant for the computation of the period, but see the section about normalization, below.		The order of the pairs is irrelevant for the computation of the period, but see the section about normalization, below.
	Once ~~they~~ are normalized, ~~these~~ signatures can be sorted and stored in a database table.		Once the signatures are normalized, they signatures can be sorted and stored in a database table.
	The original expression of eta functions can be reconstructed from them, while in the generated periods the original structure is lost.		The original expression of eta functions can be reconstructed from them, while in the generated periods the original structure is lost.

	The generation of a period from an EPSIG, the following Euler transform and the generation of the coefficients is implemented in		The generation of a period from an EPSIG, the following Euler transform and the generation of the coefficients are implemented in
	[https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/transform/EtaProductSequence.java transform/EtaProductSequence.java]		[https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/transform/EtaProductSequence.java transform/EtaProductSequence.java]
	as part of project [https://github.com/archmageirvine/joeis jOEIS] of Sean A. Irvine.		as part of the project [https://github.com/archmageirvine/joeis jOEIS] of Sean A. Irvine.
	===Common generating functions===		===Common generating functions===
	====Partition numbers====		====Partition numbers====
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	Euler transform of period 1: [1]		Euler transform of period 1: [1]
	EPSIG="[1,-1]" Q="1/24"		EPSIG="[1,-1]" Q="1/24"
	====Dedekind eta η (without the q^(1/24) factor)====		====Dedekind eta η (without the q^(1/24) factor), Ramanujan f(-q)====
	[https://oeis.org/A010815 A010815] 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1		[https://oeis.org/A010815 A010815] 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1
	G.f.: eta(q)		G.f.: eta(q)
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	EPSIG="[3,3;1,-1]" Q="-1/3" F=3		EPSIG="[3,3;1,-1]" Q="-1/3" F=3
	=== Eisenstein series===		=== Eisenstein series===
	====Ramanujan P(q) ~~= E_2~~ ====		==== E_2 = Ramanujan P(q) ====
	[https://oeis.org/A006352 A006352] 1, -24, -72, -96, -168, -144, -288, -192, -360		[https://oeis.org/A006352 A006352] 1, -24, -72, -96, -168, -144, -288, -192, -360
	a(n) = -24*sigma(n)		a(n) = -24*sigma(n)
	G.f.: 1 + 24 qderiv(eta(q))/eta(q)		G.f.: 1 + 24 qderiv(eta(q))/eta(q)
	==== Ramanujan Q(q) ~~= E_4~~ ====		==== E_4 = Ramanujan Q(q) ====
	[https://oeis.org/A004009 A004009] 1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400		[https://oeis.org/A004009 A004009] 1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400
	G.f.: (eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8		G.f.: (eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8
	EPOLY="[1,24]+256*[2,24;1,-8;2,-8]"		EPOLY="[1,24]+256*[2,24;1,-8;2,-8]"
	==== Ramanujan R(q) ~~= E_6~~ ====		==== E_6 = Ramanujan R(q) ====
	[https://oeis.org/A013973 A013973] 1, -504, -16632, -122976, -532728, -1575504, -4058208, -8471232		[https://oeis.org/A013973 A013973] 1, -504, -16632, -122976, -532728, -1575504, -4058208, -8471232
	a(0) = 1, for n >= 1: a(n) = -504*sigma_5(n)		a(0) = 1, for n >= 1: a(n) = -504*sigma_5(n)
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	G.f.: ((eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8)^2		G.f.: ((eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8)^2
	"[1,24;1,-8;2,-8]+256*[2,24;1,-8;2,-8]" squared		"[1,24;1,-8;2,-8]+256*[2,24;1,-8;2,-8]" squared
			==== E_10, E_12, E_14, etc. ====
			E_10: [https://oeis.org/A013974 A013974] a(0) = 1, for n >= 1: a(n) = -264*sigma_9(n)
			E_12: [https://oeis.org/A029828 A029828] a(0) = 691, for n >= 1: a(n) = 65520*sigma_11(n)
			E_14: [https://oeis.org/A058550 A058550] a(0) = 1, for n >= 1: a(n) = -24*sigma_13(n)
			E_16: [https://oeis.org/A029829 A029829] a(0) = 3617, for n >= 1: a(n) = 16320*sigma_15(n)
			E_18: [https://oeis.org/A279892 A279892] a(0) = 43867, for n >= 1: a(n) = -28728*sigma_17(n)
			E_20: [https://oeis.org/A029830 A029830] a(0) = 174611, for n >= 1: a(n) = 13200*sigma_19(n)
			E_22: [https://oeis.org/A279893 A279893] a(0) = 77683, for n >= 1: a(n) = -552*sigma_21(n)
			E_24: [https://oeis.org/A029831 A029831] a(0) = 236364091, for n >= 1: a(n) = 131040*sigma_23(n)
			The <code>a(0)</code> constants are given by <code>abs([https://oeis.org/A001067 A001067: Numerator of Bernoulli(2n)/(2n)])</code>, while the factors of <code>sigma_xx(n)</code> are found in <code>[https://oeis.org/A006863 A006863: Denominator of B_{2n}/(-4n</code>]
	===Leading power of q factor (pqf) ===		===Leading power of q factor (pqf) ===
	Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to <code>q^0</code>.		Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to <code>q^0</code>.
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	===Exponentiation of an EPSIG===		===Exponentiation of an EPSIG===
	An EPSIG is taken to a power ''m'' by multiplying all ''e<sub>i</sub>'' (the second components in the pairs) by ''m''. The exponent may be ''1/2'' for the square root, or any other fraction ''m<sub>n</sub>/m<sub>d</sub>'' as long as the ''q<sub>i</sub>'' are all divisible by ''m<sub>d</sub>''.		An EPSIG is taken to a power ''m'' by multiplying all ''e<sub>i</sub>'' (the second components in the pairs) by ''m''. The exponent may be ''1/2'' for the square root, or any other fraction ''m<sub>n</sub>/m<sub>d</sub>'' as long as the ''q<sub>i</sub>'' are all divisible by ''m<sub>d</sub>''.
			===Eta polynomials===
			In a number of OEIS sequences the eta products occur as components of a polynomial (cf. for example E_4, E_6, E_8 above).

Gfis: A41. E_8

2024-10-03T10:31:20Z

A41. E_8

← Older revision		Revision as of 10:31, 3 October 2024
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	[https://oeis.org/A000041 A000041] 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231		[https://oeis.org/A000041 A000041] 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231
	Euler transform of period 1: [1]		Euler transform of period 1: [1]
			EPSIG="[1,-1]" Q="1/24"
	====Dedekind eta η (without the q^(1/24) factor)====		====Dedekind eta η (without the q^(1/24) factor)====
	[https://oeis.org/A010815 A010815] 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1		[https://oeis.org/A010815 A010815] 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1
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	Euler transform of period 4: [1,-1,1,0]		Euler transform of period 4: [1,-1,1,0]
	EPSIG="[2,2;4,-1;1,-1]"		EPSIG="[2,2;4,-1;1,-1]"
	====Ramanujan f====		==== Ramanujan f ====
	[https://oeis.org/A121373 A121373] 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1		[https://oeis.org/A121373 A121373] 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1
	a(n) = (-1)^n * [https://oeis.org/A010815 A010815](n)		a(n) = (-1)^n * [https://oeis.org/A010815 A010815](n)
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	====Ramanujan a cubic AGM theta====		====Ramanujan a cubic AGM theta====
	[https://oeis.org/A004016 A004016] 1, 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6		[https://oeis.org/A004016 A004016] 1, 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6
	G.f.: theta_3(q) * theta_3(q^3) + theta_2(q) * theta_2(q^3)		G.f.: theta_3(q)theta_3(q^3) + theta_2(q)theta_2(q^3)
	(~~no EPSIG~~)		G.f.: phi(x)phi(x^3) + 4xpsi(x^2)psi(x^6)
	====Ramanujan b cubic AGM theta====		==== Ramanujan b cubic AGM theta====
	[https://oeis.org/A005928 A005928] 1, -3, 0, 6, -3, 0, 0, -6, 0, 6, 0, 0, 6, -6, 0, 0, -3		[https://oeis.org/A005928 A005928] 1, -3, 0, 6, -3, 0, 0, -6, 0, 6, 0, 0, 6, -6, 0, 0, -3
	G.f.: eta(q)^3 / (eta(q^3)		G.f.: eta(q)^3/(eta(q^3)
	Euler transform of period 3: [-3,-3,-2]		Euler transform of period 3: [-3,-3,-2]
	EPSIG="[1,3;3,-1]"		EPSIG="[1,3;3,-1]"
	====Ramanujan c cubic AGM theta====		==== Ramanujan c cubic AGM theta====
	[https://oeis.org/A005882 A005882] 3, 3, 6, 0, 6, 3, 6, 0, 3, 6, 6, 0, 6, 0, 6, 0, 9, 6, 0		[https://oeis.org/A005882 A005882] 3, 3, 6, 0, 6, 3, 6, 0, 3, 6, 6, 0, 6, 0, 6, 0, 9, 6, 0
	G.f.: 3 * eta(q^3)^3 / eta(q)		G.f.: 3*eta(q^3)^3/eta(q)
	3 * (Euler transform of period 3: [-3,-3,-2])		3 * (Euler transform of period 3: [-3,-3,-2])
	EPSIG="[3,3;1,-1]" Q="-1/3" F=3		EPSIG="[3,3;1,-1]" Q="-1/3" F=3
			=== Eisenstein series===
	====Ramanujan P(q) = E_2 ====		====Ramanujan P(q) = E_2 ====
	[https://oeis.org/A006352 A006352] 1, -24, -72, -96, -168, -144, -288, -192, -360		[https://oeis.org/A006352 A006352] 1, -24, -72, -96, -168, -144, -288, -192, -360
	a(n) = -24*sigma(n)		a(n) = -24*sigma(n)
	G.f.: 1 + 24 qderiv(eta(q))/eta(q)		G.f.: 1 + 24 qderiv(eta(q))/eta(q)
	====Ramanujan Q(q) = E_4 ====		==== Ramanujan Q(q) = E_4 ====
	[https://oeis.org/A004009 A004009] 1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400		[https://oeis.org/A004009 A004009] 1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400
	G.f.: (eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8		G.f.: (eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8
	EPOLY="[1,24]+256*[2,24;1,-8;2,-8]"		EPOLY="[1,24]+256*[2,24;1,-8;2,-8]"
	====Ramanujan R(q) = E_6 ====		==== Ramanujan R(q) = E_6 ====
	[https://oeis.org/A013973 A013973] 1, -504, -16632, -122976, -532728, -1575504, -4058208, -8471232		[https://oeis.org/A013973 A013973] 1, -504, -16632, -122976, -532728, -1575504, -4058208, -8471232
	a(0) = 1, for n >= 1: a(n) = -504*~~sigma5~~(n)		a(0) = 1, for n >= 1: a(n) = -504*sigma_5(n)
	G.f.: (eta(q)^8 + 32eta(q^4)^8) (eta(q)^16 - 512eta(q)^8eta(q^4)^8 - 8192*eta(q^4)^16) / eta(q^2)^12		G.f.: (eta(q)^8 + 32eta(q^4)^8) (eta(q)^16 - 512eta(q)^8eta(q^4)^8 - 8192*eta(q^4)^16) / eta(q^2)^12
	"[1,8]+32[4,8]" convolved with "[1,16;2,-12]-512[1,8;4,8;2,-12]-8192*[4,16;2,-12]"		"[1,8]+32[4,8]" convolved with "[1,16;2,-12]-512[1,8;4,8;2,-12]-8192*[4,16;2,-12]"
			==== E_8 ====
			[https://oeis.org/A008410 A008410] 1, 480, 61920, 1050240, 7926240, 37500480, 135480960
			a(0) = 1, for n >= 1: a(n) = 480*sigma_7(n)
			G.f.: (theta2(q)^16 + theta3(q)^16 + theta4(q)^16)/2.
			G.f.: ((eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8)^2
			"[1,24;1,-8;2,-8]+256*[2,24;1,-8;2,-8]" squared
	===Leading power of q factor (pqf) ===		===Leading power of q factor (pqf) ===
	Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to <code>q^0</code>.		Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to <code>q^0</code>.

Gfis: Ramanujan's P, Q, R

2024-10-03T10:08:39Z

Ramanujan's P, Q, R

← Older revision		Revision as of 10:08, 3 October 2024
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	[https://oeis.org/A000041 A000041] 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231		[https://oeis.org/A000041 A000041] 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231
	Euler transform of period 1: [1]		Euler transform of period 1: [1]
	====~~Dedkind~~ eta η (without the q^(1/24) factor)====		====Dedekind eta η (without the q^(1/24) factor)====
	[https://oeis.org/A010815 A010815] 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1		[https://oeis.org/A010815 A010815] 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1
	G.f.: eta(q)		G.f.: eta(q)
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	[https://oeis.org/A000122 A000122] 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2		[https://oeis.org/A000122 A000122] 1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2
	a(0) = 1, for n >= 1: a(n) = 2 if n is a square, otherwise 0.		a(0) = 1, for n >= 1: a(n) = 2 if n is a square, otherwise 0.
	G.f.: eta(q^2)^5 / (eta(q)*eta(q^4))^2		G.f.: eta(q^2)^5/(eta(q)*eta(q^4))^2
	Euler transform of period 4: [2,-3,2,-1]		Euler transform of period 4: [2,-3,2,-1]
	EPSIG="[2,5;4,-2;1,-2]"		EPSIG="[2,5;4,-2;1,-2]"
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	[https://oeis.org/A002448 A002448] 1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2		[https://oeis.org/A002448 A002448] 1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2
	a(0) = 1, for n >= 1: a(n) = 2 * (-1)^sqrt(n) if n is a square, otherwise 0.		a(0) = 1, for n >= 1: a(n) = 2 * (-1)^sqrt(n) if n is a square, otherwise 0.
	G.f.: eta(q)^2 / eta(q^2)		G.f.: eta(q)^2/eta(q^2)
	Euler transform of period 2: [2,-1]		Euler transform of period 2: [2,-1]
	EPSIG="[1,2;2,-1]"		EPSIG="[1,2;2,-1]"
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	[https://oeis.org/A010054 A010054] 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0		[https://oeis.org/A010054 A010054] 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0
	a(n) = 1 if n is a triangular number, otherwise 0.		a(n) = 1 if n is a triangular number, otherwise 0.
	G.f.: q^(-1/8) * eta(q^2)^2 / eta(q)		G.f.: q^(-1/8) * eta(q^2)^2/eta(q)
	Euler transform of period 2: [1,-1]		Euler transform of period 2: [1,-1]
	EPSIG="[2,2;1,-1]"		EPSIG="[2,2;1,-1]"
	====Ramanujan chi χ====		====Ramanujan chi χ====
	[https://oeis.org/A000700 A000700] 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5		[https://oeis.org/A000700 A000700] 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5
	G.f.: q^(1/24) * eta(q^2)^2 /(eta(q) * eta(q^4))		G.f.: q^(1/24) * eta(q^2)^2/(eta(q)*eta(q^4))
	Euler transform of period 4: [1,-1,1,0]		Euler transform of period 4: [1,-1,1,0]
	EPSIG="[2,2;4,-1;1,-1]"		EPSIG="[2,2;4,-1;1,-1]"
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	[https://oeis.org/A121373 A121373] 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1		[https://oeis.org/A121373 A121373] 1, 1, -1, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1
	a(n) = (-1)^n * [https://oeis.org/A010815 A010815](n)		a(n) = (-1)^n * [https://oeis.org/A010815 A010815](n)
	G.f.: q^(-1/24) * eta(q^2)^3 / (eta(q) * eta(q^4))		G.f.: q^(-1/24) * eta(q^2)^3/(eta(q)*eta(q^4))
	Euler transform of period 4: [1,-2,1,-1]		Euler transform of period 4: [1,-2,1,-1]
	EPSIG="[2,3;4,-1;1,-1]"		EPSIG="[2,3;4,-1;1,-1]"
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	3 * (Euler transform of period 3: [-3,-3,-2])		3 * (Euler transform of period 3: [-3,-3,-2])
	EPSIG="[3,3;1,-1]" Q="-1/3" F=3		EPSIG="[3,3;1,-1]" Q="-1/3" F=3
			====Ramanujan P(q) = E_2 ====
			[https://oeis.org/A006352 A006352] 1, -24, -72, -96, -168, -144, -288, -192, -360
			a(n) = -24*sigma(n)
			G.f.: 1 + 24 qderiv(eta(q))/eta(q)
			====Ramanujan Q(q) = E_4 ====
			[https://oeis.org/A004009 A004009] 1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400
			G.f.: (eta(q)^24 + 256eta(q^2)^24) / (eta(q)eta(q^2))^8
			EPOLY="[1,24]+256*[2,24;1,-8;2,-8]"
			====Ramanujan R(q) = E_6 ====
			[https://oeis.org/A013973 A013973] 1, -504, -16632, -122976, -532728, -1575504, -4058208, -8471232
			a(0) = 1, for n >= 1: a(n) = -504*sigma5(n)
			G.f.: (eta(q)^8 + 32eta(q^4)^8) (eta(q)^16 - 512eta(q)^8eta(q^4)^8 - 8192*eta(q^4)^16) / eta(q^2)^12
			"[1,8]+32[4,8]" convolved with "[1,16;2,-12]-512[1,8;4,8;2,-12]-8192*[4,16;2,-12]"
	===Leading power of q factor (pqf) ===		===Leading power of q factor (pqf) ===
	Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to <code>q^0</code>.		Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to <code>q^0</code>.

Gfis: R.'s tau

2024-10-01T07:26:04Z

R.'s tau

← Older revision		Revision as of 07:26, 1 October 2024
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	Euler transform of period 4: [1,-2,1,-1]		Euler transform of period 4: [1,-2,1,-1]
	EPSIG="[2,3;4,-1;1,-1]"		EPSIG="[2,3;4,-1;1,-1]"
			====Ramanujan tau τ====
			[https://oeis.org/A000594 A000594] 1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612
			G.f.: eta(q)^24
			Euler transform of period 1: [-24]
			EPSIG="[1,24]"
	====Ramanujan a cubic AGM theta====		====Ramanujan a cubic AGM theta====
	[https://oeis.org/A004016 A004016] 1, 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6		[https://oeis.org/A004016 A004016] 1, 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6

← Older revision		Revision as of 11:04, 4 October 2024
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	"[1,24;1,-8;2,-8]+256*[2,24;1,-8;2,-8]" squared		"[1,24;1,-8;2,-8]+256*[2,24;1,-8;2,-8]" squared
	==== E_10, E_12, E_14, etc. ====		==== E_10, E_12, E_14, etc. ====
	E_10: [https://oeis.org/A013974 A013974] a(0) = 1, for n >= 1: a(n) = -264*sigma_9(n)		E_10: [https://oeis.org/A013974 A013974] a(0) = 1, for n >= 1: a(n) = -264*sigma_9(n)
	E_12: [https://oeis.org/A029828 A029828] a(0) = 691, for n >= 1: a(n) = 65520*sigma_11(n)		E_12: [https://oeis.org/A029828 A029828] a(0) = 691, for n >= 1: a(n) = 65520*sigma_11(n)
	E_14: [https://oeis.org/A058550 A058550] a(0) = 1, for n >= 1: a(n) = -24*sigma_13(n)		E_14: [https://oeis.org/A058550 A058550] a(0) = 1, for n >= 1: a(n) = -24*sigma_13(n)
	E_16: [https://oeis.org/A029829 A029829] a(0) = 3617, for n >= 1: a(n) = 16320*sigma_15(n)		E_16: [https://oeis.org/A029829 A029829] a(0) = 3617, for n >= 1: a(n) = 16320*sigma_15(n)
	E_18: [https://oeis.org/A279892 A279892] a(0) = 43867, for n >= 1: a(n) = -28728*sigma_17(n)		E_18: [https://oeis.org/A279892 A279892] a(0) = 43867, for n >= 1: a(n) = -28728*sigma_17(n)
	E_20: [https://oeis.org/A029830 A029830] a(0) = 174611, for n >= 1: a(n) = 13200*sigma_19(n)		E_20: [https://oeis.org/A029830 A029830] a(0) = 174611, for n >= 1: a(n) = 13200*sigma_19(n)
	E_22: [https://oeis.org/A279893 A279893] a(0) = 77683, for n >= 1: a(n) = -552*sigma_21(n)		E_22: [https://oeis.org/A279893 A279893] a(0) = 77683, for n >= 1: a(n) = -552*sigma_21(n)
	E_24: [https://oeis.org/A029831 A029831] a(0) = 236364091, for n >= 1: a(n) = 131040*sigma_23(n)		E_24: [https://oeis.org/A029831 A029831] a(0) = 236364091, for n >= 1: a(n) = 131040*sigma_23(n)
			E_26: [https://oeis.org/A282356 A282356] a(0) = 657931, for n >= 1: a(n) = -24*sigma_25(n)
			E_28: [https://oeis.org/A282401 A282401] a(0) = 3392780147, for n >= 1: a(n) = 6960*sigma_27(n)
			E_30: [https://oeis.org/A282182 A282182] a(0) = 172316825520, for n >= 1: a(n) = -171864*sigma_29(n)
			E_32: [https://oeis.org/A282540 A282540] a(0) = 7709321041217, for n >= 1: a(n) = 32640*sigma_31(n)
	The <code>a(0)</code> constants are given by <code>abs([https://oeis.org/A001067 A001067: Numerator of Bernoulli(2n)/(2n)])</code>, while the factors of <code>sigma_xx(n)</code> are found in <code>[https://oeis.org/A006863 A006863: Denominator of B_{2n}/(-4n</code>]		The <code>a(0)</code> constants are given by <code>abs([https://oeis.org/A001067 A001067: Numerator of Bernoulli(2n)/(2n)])</code>, while the factors of <code>sigma_xx(n)</code> are found in <code>[https://oeis.org/A006863 A006863: Denominator of B_{2n}/(-4n</code>]

	===Leading power of q factor (pqf) ===		===Leading power of q factor (pqf) ===
	Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to <code>q^0</code>.		Some of the generating functions have an additional power of q factor, in order to normalize the leading coefficient to <code>q^0</code>.