A026829 Number of partitions of n into distinct parts, the least being 8.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 12, 13, 15, 17, 19, 21, 24, 26, 29, 33, 36, 40, 45, 50, 55, 62, 68, 76, 84, 93, 102, 114, 125, 138, 152, 168, 184, 204, 223, 246, 270, 297, 325, 358, 391, 429, 470

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A025147, A025154.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-8)*(i+9)/2 `if`(n<8, 0, b(n-8$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 07 2014

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[(i-8)*(i+9)/2 < n, 0, Sum[b[n - i*j, i - 1], {j, 0, Min[1, n/i]}]]]; a[n_] := If[n < 8, 0, b[n-8, n-8]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 24 2015, after Alois P. Heinz *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 8], {n, 66}]] (* Robert Price, Jun 13 2020 *)

Formula

a(n) = A025154(n-8), n>8. - R. J. Mathar, Jul 31 2008

G.f.: x^8*Product_{j>=9} (1+x^j). - R. J. Mathar, Jul 31 2008

G.f.: Sum_{k>=1} x^(k*(k + 15)/2) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 24 2020