A320733 Number of partitions of n with two sorts of part 1 which are introduced in ascending order.

1, 1, 3, 6, 13, 26, 54, 108, 219, 439, 882, 1766, 3539, 7081, 14172, 28351, 56716, 113443, 226908, 453833, 907698, 1815424, 3630893, 7261829, 14523725, 29047513, 58095121, 116190338, 232380810, 464761759, 929523710, 1859047619, 3718095507, 7436191301

Offset: 0

Author

Alois P. Heinz, Oct 20 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3322

Crossrefs

Column k=2 of A292745.

Cf. A002865, A011782, A090764.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i<2, add(
      Stirling2(n, j), j=0..2), 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, 2}], 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 *)

Formula

G.f.: ((1 - x)/(1 - 2*x)) * Product_{k>=2} 1/(1 - x^k). - Ilya Gutkovskiy, Dec 03 2019