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

0,3

Links

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

Formula

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

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 *)

Crossrefs

Column k=2 of A292745.

Cf. A002865, A011782, A090764.

Sequence in context: A081254 A125049 A267581 * A164991 A213255 A215985

Adjacent sequences: A320730 A320731 A320732 * A320734 A320735 A320736

Keyword

nonn

Author

Alois P. Heinz, Oct 20 2018

Status

approved