A320740 Number of partitions of n with nine sorts of part 1 which are introduced in ascending order.

1, 1, 3, 7, 20, 63, 233, 966, 4454, 22404, 121615, 706306, 4360204, 28452601, 195263881, 1402218667, 10482569938, 81153069799, 647261864569, 5292447172261, 44165731426846, 374675276723042, 3220404743013997, 27967105952549269, 244844437773618386

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1053

Formula

From Vaclav Kotesovec, Mar 03 2026: (Start)

a(n) ~ 9^(n-2) / (7! * QPochhammer(1/9)).

G.f.: (1 - 36*x + 539*x^2 - 4353*x^3 + 20529*x^4 - 56993*x^5 + 88620*x^6 - 67113*x^7 + 16687*x^8) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 9*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..9), 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, 9}], 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 - 36*x + 539*x^2 - 4353*x^3 + 20529*x^4 - 56993*x^5 + 88620*x^6 - 67113*x^7 + 16687*x^8) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 9*x) * Product[(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 03 2026 *)

Crossrefs

Column k=9 of A292745.

Sequence in context: A320737 A320738 A320739 * A320741 A292503 A340357

Adjacent sequences: A320737 A320738 A320739 * A320741 A320742 A320743

Keyword

nonn

Author

Alois P. Heinz, Oct 20 2018

Status

approved