A320754 Number of partitions of n with eight kinds of 1.

1, 8, 37, 129, 376, 966, 2258, 4902, 10026, 19520, 36459, 65721, 114877, 195454, 324706, 528069, 842531, 1321214, 2039553, 3103562, 4660814, 6914927, 10144558, 14728160, 21176077, 30171935, 42625765, 59741868, 83105140, 114790422, 157500479, 214739450

Offset: 0

Author

Alois P. Heinz, Oct 20 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Column k=8 of A292508.

Programs

Magma

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^8*(&*[1-x^j: j in [2..30]])))); // G. C. Greubel, Oct 27 2018

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+7)*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);

Mathematica

nmax = 50; CoefficientList[Series[1/((1-x)^7 * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 24 2018 *)

PARI

x='x+O('x^40); Vec(1/((1-x)^8*prod(j=2, 40, 1-x^j))) \\ G. C. Greubel, Oct 27 2018

Formula

G.f.: 1/(1-x)^8 * 1/Product_{j>1} (1-x^j).

Euler transform of 8,1,1,1,... .

a(n) ~ 2^(3/2) * 3^3 * n^(5/2) * exp(Pi*sqrt(2*n/3)) / Pi^7. - Vaclav Kotesovec, Oct 24 2018