A320734 Number of partitions of n with three sorts of part 1 which are introduced in ascending order.

1, 1, 3, 7, 19, 52, 151, 442, 1314, 3921, 11737, 35171, 105464, 316318, 948863, 2846461, 8539221, 25617443, 76852054, 230555794, 691666924, 2075000173, 6224999772, 18674998357, 56024993883, 168074980137, 504224938548, 1512674813304, 4538024437036, 13614073307529

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2097

Formula

From Vaclav Kotesovec, Mar 03 2026: (Start)

a(n) ~ 3^(n-2) / QPochhammer(1/3).

G.f.: (1 - 3*x + x^2) / ((1 - 3*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..3), 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, 3}], 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 - 3*x + x^2)/(1 - 3*x) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 03 2026 *)

Crossrefs

Column k=3 of A292745.

Sequence in context: A135052 A198305 A146597 * A259812 A115254 A222324

Adjacent sequences: A320731 A320732 A320733 * A320735 A320736 A320737

Keyword

nonn

Author

Alois P. Heinz, Oct 20 2018

Status

approved