A320739 Number of partitions of n with eight sorts of part 1 which are introduced in ascending order.

1, 1, 3, 7, 20, 63, 233, 966, 4454, 22403, 121570, 705150, 4337883, 28091897, 190105229, 1334705996, 9656244012, 71551215515, 540187472767, 4137336876098, 32036946594336, 250131019258467, 1965050543015106, 15509209887539395, 122829846706462146

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1112

Formula

From Vaclav Kotesovec, Mar 03 2026: (Start)

a(n) ~ 8^(n-2) / (6! * QPochhammer(1/8)).

G.f.: (1 - 28*x + 316*x^2 - 1845*x^3 + 5925*x^4 - 10190*x^5 + 8249*x^6 - 2119*x^7) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 8*x) * Product_{k>=1} (1 - x^k)). (End)

Maple

b:= proc(n, i) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..8), add(b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i < 2, Sum[StirlingS2[n, j], {j, 0, 8}], Sum[b[n - i j, i - 1], {j, 0, n/i}]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)
(* or *)
nmax = 40; CoefficientList[Series[(1 - 28*x + 316*x^2 - 1845*x^3 + 5925*x^4 - 10190*x^5 + 8249*x^6 - 2119*x^7) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 8*x) * Product[(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 03 2026 *)

Crossrefs

Column k=8 of A292745.

Sequence in context: A320736 A320737 A320738 * A320740 A320741 A292503

Adjacent sequences: A320736 A320737 A320738 * A320740 A320741 A320742

Keyword

nonn

Author

Alois P. Heinz, Oct 20 2018

Status

approved