OEIS/Infinite Products - Revision history

Gfis: MMA

2026-03-09T14:59:29Z

MMA

← Older revision		Revision as of 14:59, 9 March 2026
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	=== List of common products ===		=== List of common products ===
	Maple: <code>seq(coeff(series(mul((1-3*x^k), k = 1..n), x, n+1), x, n), n=0..15);</code>		Maple: <code>seq(coeff(series(mul((1-3*x^k), k = 1..n), x, n+1), x, n), n=0..15);</code>

			Mathematica: [[oeis:A000700\|A000700]] <code>CoefficientList[ Series[ Product[1 + x^(2k + 1), {k, 0, 75}], {x, 0, 70}], x]; also: 1/Product_{i>=1} (1 + (-x)^i)</code>

	====(1+m*x^k)====		====(1+m*x^k)====
	* [https://oeis.org/A000009 A000009] Product_{k>=1} (1+1*x^k)		* [https://oeis.org/A000009 A000009] Product_{k>=1} (1+1*x^k)

Gfis: RPT

2026-03-09T08:27:31Z

RPT

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 # If we set f(n) = - n ([https://oeis.org/A001478 A001478]) and g(n) = n ([https://oeis.org/A000027 A000027]), we get A266964.
 # With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n).
 === List of common products ===
 Maple: <code>seq(coeff(series(mul((1-3*x^k), k = 1..n), x, n+1), x, n), n=0..15);</code>

Gfis: /* List of common products */

2023-12-15T13:31:03Z

List of common products

← Older revision		Revision as of 13:31, 15 December 2023
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	# With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n).		# With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n).
	=== List of common products ===		=== List of common products ===
	Maple:		Maple: <code>seq(coeff(series(mul((1-3*x^k), k = 1..n), x, n+1), x, n), n=0..15);</code>
	seq(coeff(series(mul((1-3*x^k), k = 1..n), x, n+1), x, n), n=0..15);		====(1+m*x^k)====
	* [https://oeis.org/A000009 A000009] Product_{k>=1} (1+1*x^k)		* [https://oeis.org/A000009 A000009] Product_{k>=1} (1+1*x^k)
	* [https://oeis.org/A032302 A032302] Product_{k>=1} (1+2*x^k)		* [https://oeis.org/A032302 A032302] Product_{k>=1} (1+2*x^k)
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	* [https://oeis.org/A246936 A246936] Product_{k>=1} (1-4*x^k)^(-1)		* [https://oeis.org/A246936 A246936] Product_{k>=1} (1-4*x^k)^(-1)
	* [https://oeis.org/A246937 A246937] Product_{k>=1} (1-5*x^k)^(-1)		* [https://oeis.org/A246937 A246937] Product_{k>=1} (1-5*x^k)^(-1)
			====(1+mkx^k)====
	* [https://oeis.org/A265951 A006906] Product_{k>=1} (1-1kx^k)^(-1)		* [https://oeis.org/A265951 A006906] Product_{k>=1} (1-1kx^k)^(-1)
	* [https://oeis.org/A265951 A265951] Product_{k>=1} (1-2kx^k)^(-1)		* [https://oeis.org/A265951 A265951] Product_{k>=1} (1-2kx^k)^(-1)

Gfis: 1+-mkx^k

2023-12-15T13:29:15Z

1+-m*k*x^k

← Older revision		Revision as of 13:29, 15 December 2023
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	* [https://oeis.org/A246936 A246936] Product_{k>=1} (1-4*x^k)^(-1)		* [https://oeis.org/A246936 A246936] Product_{k>=1} (1-4*x^k)^(-1)
	* [https://oeis.org/A246937 A246937] Product_{k>=1} (1-5*x^k)^(-1)		* [https://oeis.org/A246937 A246937] Product_{k>=1} (1-5*x^k)^(-1)

			* [https://oeis.org/A265951 A006906] Product_{k>=1} (1-1kx^k)^(-1)
			* [https://oeis.org/A265951 A265951] Product_{k>=1} (1-2kx^k)^(-1)
			* [https://oeis.org/A265951 A265974] Product_{k>=1} (1-3kx^k)^(-1)
			* [https://oeis.org/A265951 A265975] Product_{k>=1} (1-4kx^k)^(-1)
			* [https://oeis.org/A265951 A265976] Product_{k>=1} (1-5kx^k)^(-1)
			* [https://oeis.org/A032309 A022629] Product_{k>=1} (1+1kx^k)
			* [https://oeis.org/A032309 A022693] Product_{k>=1} (1+1kx^k)^(-1)
			* [https://oeis.org/A032309 A032309] Product_{k>=1} (1+2kx^k)

Gfis: List of common products

2023-12-15T13:12:09Z

List of common products

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 # If we set f(n) = - n ([https://oeis.org/A001478 A001478]) and g(n) = n ([https://oeis.org/A000027 A000027]), we get A266964.
 # With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n).
-=== Interesting sequences ===
+=== List of common products ===
-* [https://oeis.org/A000081 A000081] Euler transform of itself, shifted by 1
+Maple:

imported>Gfis: Transforms

2020-12-31T18:19:03Z

Transforms

← Older revision		Revision as of 18:19, 31 December 2020
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			=== OEIS collection of transforms ===
			* [https://oeis.org/wiki/Sequence_transforms Sequence transforms]: [https://oeis.org/transforms.txt Maple], [https://oeis.org/seqtranslib.txt Mathematica]
			weighout b from a in 1+SUM b_n x^n = PI (1+x^n)^a_n
			weighouti a from b in 1+SUM b_n x^n = PI (1+x^n)^a_n
			weighini a from b in 1+SUM b_n x^n = PI (1+x^a_n)
			weigh2out b from a in 1+SUM b_n x^n = PI (x^-n+1+x^n)^a_n
			weigh2outi a from b in 1+SUM b_n x^n = PI (x^-n+1+x^n)^a_n
			weigh2in b from a in 1+SUM b_n x^n = PI (x^-a_n+1+x^a_n)
			weigh2ini a from b in 1+SUM b_n x^n = PI (x^-a_n+1+x^a_n)
	=== Programs ===		=== Programs ===
	[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.		[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.

imported>Gfis: /* Infinite Products in your CAS */ Maple2

2020-12-09T08:22:16Z

Infinite Products in your CAS: Maple2

← Older revision		Revision as of 08:22, 9 December 2020
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	=== ~~Infinite Products in your CAS~~ ===		=== Programs ===
	[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.		[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.
	(Maple)		(Maple)
	g := -1 + 1/product(1 - x^(j^4), j=1..10):		g := -1 + 1/product(1 - x^(j^4), j=1..10):
	gser := series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102);		gser := series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102);
			(Maple, [https://oeis.org/A303350 A303350])
			seq(coeff(series(mul((1+4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40);
	(Mathematica)		(Mathematica)
	g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}];		g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}];
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	{a(n) = polcoeff( 1/prod(m=1, n, 1 - (2^m+1)x^m +xO(x^n)), n)}		{a(n) = polcoeff( 1/prod(m=1, n, 1 - (2^m+1)x^m +xO(x^n)), n)}
	for(n=0, 30, print1(a(n), ","))		for(n=0, 30, print1(a(n), ","))

	=== Generalized Euler transform ===		=== Generalized Euler transform ===
	* defined by [https://oeis.org/wiki/User:Seiichi_Manyama Seiichi Manyama] in [https://oeis.org/A266964 A266964]		* defined by [https://oeis.org/wiki/User:Seiichi_Manyama Seiichi Manyama] in [https://oeis.org/A266964 A266964]

imported>Gfis: PARI

2020-12-07T21:00:26Z

PARI

← Older revision		Revision as of 21:00, 7 December 2020
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	=== Infinite Products in your CAS ===		=== Infinite Products in your CAS ===
	[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.		[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.
	* Maple		(Maple)
	g := -1 + 1/product(1 - x^(j^4), j=1..10):		g := -1 + 1/product(1 - x^(j^4), j=1..10):
	gser := series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102);		gser := series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102);
	* Mathematica		(Mathematica)
	g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}];		g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}];
	gser = Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}]		gser = Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}]
			(PARI, [https://oeis.org/A322199 A322199])
			{a(n) = polcoeff( 1/prod(m=1, n, 1 - (2^m+1)x^m +xO(x^n)), n)}
			for(n=0, 30, print1(a(n), ","))
	=== Generalized Euler transform ===		=== Generalized Euler transform ===
	* defined by [https://oeis.org/wiki/User:Seiichi_Manyama Seiichi Manyama] in [https://oeis.org/A266964 A266964]		* defined by [https://oeis.org/wiki/User:Seiichi_Manyama Seiichi Manyama] in [https://oeis.org/A266964 A266964]
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	# If we set f(n) = - n ([https://oeis.org/A001478 A001478]) and g(n) = n ([https://oeis.org/A000027 A000027]), we get A266964.		# If we set f(n) = - n ([https://oeis.org/A001478 A001478]) and g(n) = n ([https://oeis.org/A000027 A000027]), we get A266964.
	# With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n).		# With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n).
			=== Interesting sequences ===
			* [https://oeis.org/A000081 A000081] Euler transform of itself, shifted by 1

imported>Gfis: bracket

2020-12-05T09:01:41Z

bracket

imported>Gfis: Initial

2020-12-05T08:58:16Z

Initial

New page

=== Infinite Products in your CAS ===
[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.
* Maple
g:=-1+1/product(1-x^(j^4), j=1..10):
gser:=series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102);
* Mathematica
g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}];
gser = Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}]
=== Generalized Euler transform ===
* defined by Seiichi Manyama in [https://oeis.org/A266964 A266964]
Product_{k>0} (1 - g(k)*x^k)^(- f(k)) = a(0) + a(1)*x + a(2)*x^2 + ...
# If we set g(n) = 1, we get the usual https://oeis.org/wiki/Euler_transform Euler transform].
# If we set f(n) = - h(n) and g(n) = -1, we get the weighout transform (cf. [https://oeis.org/A026007 A026007]).
# If we set f(n) = - n ([https://oeis.org/A001478 A001478]) and g(n) = n ([https://oeis.org/A000027 A000027]), we get A266964.
# With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n).

← Older revision		Revision as of 09:01, 5 December 2020
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	[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.		[https://oeis.org/A046042 A046042] Number of partitions of n into fourth powers.
	* Maple		* Maple
	g:=-1+1/product(1-x^(j^4), j=1..10):		g := -1 + 1/product(1 - x^(j^4), j=1..10):
	gser:=series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102);		gser := series(g, x=0, 105): seq(coeff(gser, x, n), n=1..102);
	* Mathematica		* Mathematica
	g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}];		g = -1 + 1/Product[1 - x^(j^4), {j, 1, 10}];
	gser = Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}]		gser = Series[g, {x, 0, 105}]; Table[Coefficient[gser, x, n], {n, 1, 102}]
	=== Generalized Euler transform ===		=== Generalized Euler transform ===
	* defined by Seiichi Manyama in [https://oeis.org/A266964 A266964]		* defined by [https://oeis.org/wiki/User:Seiichi_Manyama Seiichi Manyama] in [https://oeis.org/A266964 A266964]
	Product_{k>0} (1 - g(k)x^k)^(- f(k)) = a(0) + a(1)x + a(2)*x^2 + ...		Product_{k>0} (1 - g(k)x^k)^(- f(k)) = a(0) + a(1)x + a(2)*x^2 + ...
	# If we set g(n) = 1, we get the usual https://oeis.org/wiki/Euler_transform Euler transform].		# If we set g(n) = 1, we get the usual '''[https://oeis.org/wiki/Euler_transform Euler transform]'''.
	# If we set f(n) = - h(n) and g(n) = -1, we get the weighout transform (cf. [https://oeis.org/A026007 A026007]).		# If we set f(n) = - h(n) and g(n) = -1, we get the weighout transform (cf. [https://oeis.org/A026007 A026007]).
	# If we set f(n) = - n ([https://oeis.org/A001478 A001478]) and g(n) = n ([https://oeis.org/A000027 A000027]), we get A266964.		# If we set f(n) = - n ([https://oeis.org/A001478 A001478]) and g(n) = n ([https://oeis.org/A000027 A000027]), we get A266964.
	# With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n).		# With the default f(n) = g(n) = [https://oeis.org/A000012 A000012] (all 1's), we get [https://oeis.org/A000041 A000041] (number of partitions of n).