A022581 - OEIS

%I #39 Sep 16 2025 17:16:03

%S 1,16,136,832,4132,17696,67712,236928,770442,2355824,6834240,18940480,

%T 50424536,129535968,322288128,779022208,1834203955,4216133616,

%U 9479688992,20884408704,45148577668,95902505120,200394848512,412350614016,836328261438,1673337795840,3305364030464,6450386567104,12443955363352,23745951691328,44844655553536,83856163515776,155331420821337

%N Expansion of Product_{m>=1} (1+x^m)^16.

%H Seiichi Manyama, <a href="/A022581/b022581.txt">Table of n, a(n) for n = 0..10000</a>

%F Expansion of q^(-2/3)(eta(q^2)/eta(q))^16 in powers of q. - _Michael Somos_, Jun 06 2005

%F Euler transform of period 2 sequence [16, 0, ...]. - _Michael Somos_, Jun 06 2005

%F G.f.: G(x) = (Prod_{k>0} 1+x^k)^16.

%F Let P(x) = prod(n>=1, (1+x^n)) (the g.f. for partitions into distinct parts, A000009). Then P(x^2)^8 + 16*x*P(x^2)^16*P(x)^8 = P(x)^16 (cf. A022581). - _Joerg Arndt_, Jul 12 2009

%F a(n) ~ exp(4 * Pi * sqrt(n/3)) / (256 * sqrt(2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Mar 05 2015

%F a(0) = 1, a(n) = (16/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 03 2017

%F Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/4) * exp(2*Pi/3) = A388307. - _Simon Plouffe_, Sep 15 2025

%t nmax=50; CoefficientList[Series[Product[(1+q^m)^16,{m,1,nmax}],{q,0,nmax}],q] (* _Vaclav Kotesovec_, Mar 05 2015 *)

%t s = (QPochhammer[-1, q]/2)^16 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)

%o (PARI) q='q+O('q^66); gf=(eta(q^2)/eta(q))^16; Vec(gf) \\ _Joerg Arndt_, Jul 06 2011

%o (PARI) m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^16)) \\ _G. C. Greubel_, Feb 25 2018

%o (Magma) Coefficients(&*[(1+x^m)^16:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 25 2018

%Y Column k=16 of A286335.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jun 14 1998