A026831 Number of partitions of n into distinct parts, the least being 10.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 11, 13, 14, 16, 18, 20, 22, 25, 27, 30, 33, 36, 40, 44, 48, 53, 59, 64, 71, 78, 86, 94, 104, 113, 125, 136, 149, 163, 179, 194, 213, 232, 254, 276, 302

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Examples

			Say n = 11. There is no way to partition 11 into n distinct parts if one of the least parts is 10 since 11 = 10 + x where x >= 10 has no solutions. Hence a(11) = 0.
		

Links

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

Crossrefs

Cf. A025147, A025155, A096740.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-10)*(i+11)/2 `if`(n<10, 0, b(n-10$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-10)*(i+11)/2 < n, 0, Sum[b[n-i*j, i-1], {j, 0, Min[1, n/i]}]]]; a[n_] := If[n<10, 0, b[n-10, n-10]]; 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[#] == # &], 10], {n, 66}]] (* Robert Price, Jun 13 2020 *)

Formula

a(n) = A096740(n-10), n>10. - R. J. Mathar, Jul 31 2008

G.f.: x^10*Product_{j>=11} (1+x^j). - R. J. Mathar, Jul 31 2008

G.f.: Sum_{k>=1} x^(k*(k + 19)/2) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020