A343352 Expansion of Product_{k>=1} 1 / (1 - x^k)^(7^(k-1)).

1, 1, 8, 57, 428, 3172, 23689, 176324, 1312550, 9757798, 72480269, 537854094, 3987751860, 29540543908, 218652961074, 1617159619805, 11951595353413, 88264810625245, 651404299886762, 4804261815210433, 35410065096578748, 260832137791524693, 1920169120639498017, 14127684273966098698

Offset: 0

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

• nonn

Crossrefs

Cf. A034691, A104460, A144071, A343349, A343350, A343351, A343353, A343354, A343355.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*7^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2021

Mathematica

nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(7^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 7^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

Formula

a(n) ~ exp(2*sqrt(n/7) - 1/14 + c/7) * 7^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (7^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021