A022581 Expansion of Product_{m>=1} (1+x^m)^16.

1, 16, 136, 832, 4132, 17696, 67712, 236928, 770442, 2355824, 6834240, 18940480, 50424536, 129535968, 322288128, 779022208, 1834203955, 4216133616, 9479688992, 20884408704, 45148577668, 95902505120, 200394848512, 412350614016, 836328261438, 1673337795840, 3305364030464, 6450386567104, 12443955363352, 23745951691328, 44844655553536, 83856163515776, 155331420821337

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

Expansion of q^(-2/3)(eta(q^2)/eta(q))^16 in powers of q. - Michael Somos, Jun 06 2005

Euler transform of period 2 sequence [16, 0, ...]. - Michael Somos, Jun 06 2005

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

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

a(n) ~ exp(4 * Pi * sqrt(n/3)) / (256 * sqrt(2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

a(0) = 1, a(n) = (16/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

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

Mathematica

nmax=50; CoefficientList[Series[Product[(1+q^m)^16, {m, 1, nmax}], {q, 0, nmax}], q] (* Vaclav Kotesovec, Mar 05 2015 *)
s = (QPochhammer[-1, q]/2)^16 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

Prog

(PARI) q='q+O('q^66); gf=(eta(q^2)/eta(q))^16; Vec(gf) \\ Joerg Arndt, Jul 06 2011
(PARI) m=50; q='q+O('q^m); Vec(prod(n=1, m, (1+q^n)^16)) \\ G. C. Greubel, Feb 25 2018
(Magma) Coefficients(&*[(1+x^m)^16:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

Crossrefs

Column k=16 of A286335.

Sequence in context: A290896 A223031 A341227 * A383913 A387846 A278283

Adjacent sequences: A022578 A022579 A022580 * A022582 A022583 A022584

Keyword

nonn

Author

N. J. A. Sloane, Jun 14 1998

Status

approved