A343364 - OEIS

%I #10 Apr 13 2021 04:45:05

%S 1,1,7,56,413,3108,23163,172711,1285256,9556603,70980000,526711507,

%T 3904946864,28926003505,214095348671,1583389916081,11701578676851,

%U 86415267247743,637732279701496,4703270177738076,34664585073280204,255332979654402524,1879629724498860397,13829015594546304600

%N Expansion of Product_{k>=1} (1 + x^k)^(7^(k-1)).

%F 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 / (j * (7^(j-1) - 1)). - _Vaclav Kotesovec_, Apr 13 2021

%p h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(h(n-i*j, i-1)*binomial(7^(i-1), j), j=0..n/i)))

%p     end:

%p a:= n-> h(n$2):

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Apr 12 2021

%t nmax = 23; CoefficientList[Series[Product[(1 + x^k)^(7^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 7^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

%o (PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(7^(k-1))))} \\ _Andrew Howroyd_, Apr 12 2021

%Y Cf. A098407, A292841, A343352, A343360, A343361, A343362, A343363, A343365, A343366.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Apr 12 2021