A343364 Expansion of Product_{k>=1} (1 + x^k)^(7^(k-1)).

1, 1, 7, 56, 413, 3108, 23163, 172711, 1285256, 9556603, 70980000, 526711507, 3904946864, 28926003505, 214095348671, 1583389916081, 11701578676851, 86415267247743, 637732279701496, 4703270177738076, 34664585073280204, 255332979654402524, 1879629724498860397, 13829015594546304600

Offset

0,3

Links

Table of n, a(n) for 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 / (j * (7^(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(7^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2021

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A104896 A246939 A122996 * A092318 A092315 A229248

Adjacent sequences: A343361 A343362 A343363 * A343365 A343366 A343367

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 12 2021

Status

approved