A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).

1, 1, 5, 21, 95, 415, 1851, 8155, 36030, 158510, 696502, 3052966, 13359230, 58346206, 254405630, 1107479694, 4813850699, 20894227355, 90567536543, 392066476815, 1695180397145, 7320927664713, 31581573600685, 136094434672509, 585876330191950, 2519701493092958

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Formula

a(n) ~ exp(sqrt(n) - 1/8 + c/4) * 2^(2*n - 3/2) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (4^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*4^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 12 2021

Mathematica

nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^(4^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

Crossrefs

Cf. A034691, A104460, A144068, A343350, A343351, A343352, A343353, A343354, A343355.

Sequence in context: A178876 A202513 A361775 * A159289 A201869 A017968

Adjacent sequences: A343346 A343347 A343348 * A343350 A343351 A343352

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 12 2021

Status

approved