A022570 Expansion of Product_{m>=1} (1+x^m)^5.

1, 5, 15, 40, 95, 206, 425, 835, 1575, 2880, 5121, 8885, 15095, 25165, 41240, 66562, 105945, 166480, 258560, 397235, 604162, 910325, 1359680, 2014235, 2961000, 4321283, 6263360, 9019555, 12908945, 18367805, 25990149, 36581200, 51228175, 71393555, 99037095, 136775685, 188091960

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.

Crossrefs

Cf. A000009.

Column k=5 of A286335.

Programs

Magma

Coefficients(&*[(1+x^m)^5:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

Mathematica

nmax=50; CoefficientList[Series[Product[(1+q^m)^5,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)

PARI

x='x+O('x^51); Vec(prod(m=1, 50, (1 + x^m)^5)) \\ Indranil Ghosh, Apr 03 2017

Formula

a(n) ~ (5/3)^(1/4) * exp(Pi * sqrt(5*n/3)) / (16 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

G.f.: exp(5*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018