A319919 Expansion of Product_{k>=1} (1 + x^k)^(2^k-1).

1, 1, 3, 10, 25, 70, 182, 476, 1220, 3122, 7883, 19794, 49340, 122237, 301114, 737923, 1799597, 4369204, 10563800, 25441377, 61048713, 145988775, 347981713, 826921992, 1959363778, 4629903905, 10911757432, 25652950459, 60165831361, 140792215037, 328750398275, 766041930160, 1781452975346

Offset: 0

Author

Ilya Gutkovskiy, Oct 01 2018

Keywords

• nonn

Comments

Convolution of A081362 and A102866.

Weigh transform of A000225.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000225, A007070, A081362, A098407, A102866, A319918.

Programs

Maple

a:=series(mul((1+x^k)^(2^k-1),k=1..100),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^k/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

Formula

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)*(1 - 2*x^k))).

a(n) ~ c * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A079555 * sqrt(Pi) * n^(3/4)), where c = exp(Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1))) = 0.6602994483152065685... - Vaclav Kotesovec, Sep 15 2021