A319918 Expansion of Product_{k>=1} 1/(1 - x^k)^(2^k-1).

1, 1, 4, 11, 32, 84, 230, 597, 1567, 4020, 10286, 25994, 65387, 163065, 404617, 997687, 2448220, 5977334, 14530835, 35173496, 84814982, 203760809, 487845377, 1164191563, 2769721073, 6570218773, 15542642042, 36671354125, 86306246887, 202637312099, 474684979292, 1109539437382

Offset

0,3

Comments

Convolution of A010815 and A034899.

Euler transform of A000225.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3171

N. J. A. Sloane, Transforms

Formula

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

a(n) ~ A247003^2 * exp(2*sqrt(n) - 1/2) * 2^(n-1) / (A065446 * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 15 2021

Maple

a:=series(mul(1/(1-x^k)^(2^k-1), k=1..100), x=0, 32): seq(coeff(a, x, n), n=0..31); # Paolo P. Lava, Apr 02 2019
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
       d*(2^d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Aug 13 2021

Mathematica

nmax = 31; CoefficientList[Series[Product[1/(1 - x^k)^(2^k - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 31; CoefficientList[Series[Exp[Sum[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[d (2^d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 31}]

Crossrefs

Cf. A000225, A007070, A010815, A034691, A034899, A319919.

Sequence in context: A353425 A155962 A027153 * A034754 A345029 A268744

Adjacent sequences: A319915 A319916 A319917 * A319919 A319920 A319921

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 01 2018

Status

approved