OEIS/Infinite Products: Difference between revisions
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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) | |||
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) | |||
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 +x*O(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 | |||
Revision as of 21:00, 7 December 2020
Infinite Products in your CAS
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}]
(PARI, A322199)
{a(n) = polcoeff( 1/prod(m=1, n, 1 - (2^m+1)*x^m +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ","))
Generalized Euler transform
- defined by Seiichi Manyama in 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 Euler transform.
- If we set f(n) = - h(n) and g(n) = -1, we get the weighout transform (cf. A026007).
- If we set f(n) = - n (A001478) and g(n) = n (A000027), we get A266964.
- With the default f(n) = g(n) = A000012 (all 1's), we get A000041 (number of partitions of n).
Interesting sequences
- A000081 Euler transform of itself, shifted by 1