OEIS/Infinite Products: Difference between revisions
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imported>Gfis PARI |
imported>Gfis →Infinite Products in your CAS: Maple2 |
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=== | === 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 +x*O(x^n)), n)} | {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), ",")) | 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] |
Revision as of 08:22, 9 December 2020
Programs
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); (Maple, A303350) seq(coeff(series(mul((1+4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40); (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