A343362 Expansion of Product_{k>=1} (1 + x^k)^(5^(k-1)).
1, 1, 5, 30, 160, 885, 4810, 26185, 142005, 769305, 4159301, 22455876, 121057525, 651737675, 3504241650, 18818709130, 100945053055, 540885242825, 2895159035375, 15481318817450, 82704855762375, 441427664993275, 2354020475714775, 12542918682786300, 66778882780674975
Offset: 0
Keywords
Programs
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Maple
h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(h(n-i*j, i-1)*binomial(5^(i-1), j), j=0..n/i))) end: a:= n-> h(n$2): seq(a(n), n=0..24); # Alois P. Heinz, Apr 12 2021 -
Mathematica
nmax = 24; CoefficientList[Series[Product[(1 + x^k)^(5^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 5^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}] -
PARI
seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(5^(k-1))))} \\ Andrew Howroyd, Apr 12 2021
Formula
a(n) ~ exp(2*sqrt(n/5) - 1/10 - c/5) * 5^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (5^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021