A022571 Expansion of Product_{m>=1} (1+x^m)^6.

1, 6, 21, 62, 162, 384, 855, 1806, 3648, 7110, 13434, 24702, 44361, 78006, 134592, 228302, 381300, 627840, 1020394, 1638528, 2601849, 4088780, 6363354, 9813504, 15005458, 22760262, 34261248, 51204222, 76005906, 112092438, 164296989, 239404860, 346898496, 499971968, 716906394

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", New York, Gordon and Breach Science Publishers, 1986-1992, p. 755, Eq. 6.2.2.2. MR0874986 (88f:00013)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000009.

Column k=6 of A286335.

Programs

Magma

Coefficients(&*[(1+x^m)^6: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)^6,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)

PARI

a(n)=if(n<0, 0, polcoeff( prod(k=1,n, 1+x^k, 1+x*O(x^n))^6, n)) /* Michael Somos, Jul 09 2005 */

Formula

Euler transform of period 2 sequence [6, 0, ...]. - Michael Somos, Jul 09 2005

Expansion of q^(-1/4)(eta(q^2)/eta(q))^6 in powers of q. - Michael Somos, Jul 09 2005

Expansion of q^(-1/4)(1/2)k^(1/2)/k' in powers of q. - Michael Somos, Jul 09 2005

Given g.f. A(x), then B(x)=(x*A(x^4))^4 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(4096uv+48u+1)v-u^2 . - Michael Somos, Jul 09 2005

Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(u^2-v^2)^2 -uv(1+8uv)^2 . - Michael Somos, Jul 09 2005

G.f.: Product_{k>0} (1+x^k)^6.

a(n) ~ exp(Pi * sqrt(2*n)) / (16 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

G.f.: exp(6*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018