A343365 Expansion of Product_{k>=1} (1 + x^k)^(8^(k-1)).

1, 1, 8, 72, 604, 5148, 43544, 368408, 3112262, 26273542, 221605240, 1867736120, 15730022540, 132385106956, 1113413229000, 9358220560136, 78606905495809, 659886123312449, 5536404584185376, 46424396382193376, 389074608184431328, 3259085506224931424, 27286163457927575200

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

a(n) ~ exp(sqrt(n/2) - 1/16 - c/8) * 2^(3*n - 7/4) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (8^(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(8^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2021

Mathematica

nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(8^(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 8^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

Prog

(PARI) seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(8^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

Crossrefs

Cf. A098407, A292842, A343353, A343360, A343361, A343362, A343363, A343364, A343366.

Sequence in context: A158798 A229249 A242160 * A062541 A057091 A156566

Adjacent sequences: A343362 A343363 A343364 * A343366 A343367 A343368

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 12 2021

Status

approved