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

0,2

Links

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

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

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 *)

Prog

(PARI) x='x+O('x^40); Vec(1/((1-x)^8*prod(j=2, 40, 1-x^j))) \\ G. C. Greubel, Oct 27 2018
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^8*(&*[1-x^j: j in [2..30]])))); // G. C. Greubel, Oct 27 2018

Crossrefs

Column k=8 of A292508.

Sequence in context: A052387 A001780 A258476 * A053296 A055799 A035038

Adjacent sequences: A320751 A320752 A320753 * A320755 A320756 A320757

Keyword

nonn

Author

Alois P. Heinz, Oct 20 2018

Status

approved