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

0,3

Links

Table of n, a(n) for n=0..24.

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

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}]

Prog

(PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(5^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

Crossrefs

Cf. A098407, A292839, A343350, A343360, A343361, A343363, A343364, A343365, A343366.

Sequence in context: A122995 A254944 A003731 * A055838 A318591 A094972

Adjacent sequences: A343359 A343360 A343361 * A343363 A343364 A343365

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 12 2021

Status

approved