A022571 - OEIS

%I #32 Sep 08 2022 08:44:46

%S 1,6,21,62,162,384,855,1806,3648,7110,13434,24702,44361,78006,134592,

%T 228302,381300,627840,1020394,1638528,2601849,4088780,6363354,9813504,

%U 15005458,22760262,34261248,51204222,76005906,112092438,164296989,239404860,346898496,499971968,716906394

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

%D 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)

%H Seiichi Manyama, <a href="/A022571/b022571.txt">Table of n, a(n) for n = 0..1000</a>

%F Euler transform of period 2 sequence [6, 0, ...]. - _Michael Somos_, Jul 09 2005

%F Expansion of q^(-1/4)(eta(q^2)/eta(q))^6 in powers of q. - _Michael Somos_, Jul 09 2005

%F Expansion of q^(-1/4)(1/2)k^(1/2)/k' in powers of q. - _Michael Somos_, Jul 09 2005

%F 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

%F 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

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

%F a(n) ~ exp(Pi * sqrt(2*n)) / (16 * 2^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Mar 05 2015

%F a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 03 2017

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

%t nmax=50; CoefficientList[Series[Product[(1+q^m)^6,{m,1,nmax}],{q,0,nmax}],q] (* _Vaclav Kotesovec_, Mar 05 2015 *)

%o (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 */

%o (Magma) Coefficients(&*[(1+x^m)^6:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 26 2018

%Y Cf. A000009.

%Y Column k=6 of A286335.

%K nonn

%O 0,2

%A _N. J. A. Sloane_