A320737 Number of partitions of n with six sorts of part 1 which are introduced in ascending order.

1, 1, 3, 7, 20, 63, 233, 965, 4425, 21904, 114910, 628754, 3544272, 20393306, 118986963, 700768255, 4152987416, 24714368292, 147480695339, 881688073414, 5277421580515, 31613933962624, 189481916086717, 1136086826214117, 6813308511956936, 40867019987219945

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1288

Formula

From Vaclav Kotesovec, Mar 03 2026: (Start)

a(n) ~ 6^(n-3) / (4*QPochhammer(1/6)).

G.f.: (1 - 15*x + 81*x^2 - 192*x^3 + 189*x^4 - 53*x^5) / ((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*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..6), 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, 6}], 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 - 15*x + 81*x^2 - 192*x^3 + 189*x^4 - 53*x^5)/((1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x) * Product[(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 03 2026 *)

Crossrefs

Column k=6 of A292745.

Sequence in context: A320735 A176697 A320736 * A320738 A320739 A320740

Adjacent sequences: A320734 A320735 A320736 * A320738 A320739 A320740

Keyword

nonn

Author

Alois P. Heinz, Oct 20 2018

Status

approved