A343363 Expansion of Product_{k>=1} (1 + x^k)^(6^(k-1)).

1, 1, 6, 42, 267, 1743, 11234, 72470, 466251, 2996883, 19234836, 123315828, 789682546, 5051601010, 32282443044, 206104519572, 1314652656453, 8378283675645, 53350205335626, 339445117302366, 2158091256282273, 13710402587540469, 87040883294333382, 552205562345916570

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..1000

Formula

a(n) ~ exp(sqrt(2*n/3) - 1/12 - c/6) * 6^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (6^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

Maple

N:= 100: # for a(0)..a(N)
G:= mul((1+x^k)^(6^(k-1)), k=1..N):
S:= series(G, x, N+1):
seq(coeff(S, x, k), k=0..N); # Robert Israel, Apr 12 2021

Mathematica

nmax = 23; CoefficientList[Series[Product[(1 + x^k)^(6^(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 6^(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))^(6^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

Crossrefs

Cf. A098407, A292840, A343351, A343360, A343361, A343362, A343364, A343365, A343366.

Sequence in context: A158797 A218060 A283328 * A331706 A074429 A062310

Adjacent sequences: A343360 A343361 A343362 * A343364 A343365 A343366

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 12 2021

Status

approved