A343364 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

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

(list; graph; refs; listen; history; text; internal format)

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