A025151 Number of partitions of n into distinct parts >= 6.

1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 20, 23, 26, 29, 34, 38, 43, 49, 55, 62, 70, 79, 88, 100, 111, 125, 140, 157, 174, 196, 217, 243, 270, 301, 333, 372, 411, 457, 506, 561, 619, 687, 757, 837, 924, 1019, 1122, 1238, 1361, 1498

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.

Crossrefs

Cf. A025147.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-5)*(i+6)/2 b(n$2):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 07 2014

Mathematica

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 6 &]; Table[d[n], {n, 16}] (* strict partitions, parts >= 6 *)
Table[Length[d[n]], {n, 40}] (* A025151 for n >= 1 *)
(* Clark Kimberling, Mar 07 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[(i - 5)(i + 6)/2 < n, 0, Sum[b[n - i j, i - 1], {j, 0, Min[1, n/i]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

Formula

a(n) = A026826(n+5). - R. J. Mathar, Jul 31 2008

G.f.: Product_{j>=6} (1+x^j). - R. J. Mathar, Jul 31 2008

G.f.: Sum_{k>=0} x^(k*(k + 11)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 24 2020