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
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..3171
- N. J. A. Sloane, Transforms
Programs
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Maple
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}]
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
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