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

0,34

Links

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

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

Example

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.

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-10)*(i+11)/2<n, 0,
       add(b(n-i*j, i-1), j=0..min(1, n/i))))
    end:
a:= n-> `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 *)

Crossrefs

Cf. A025147, A025155, A096740.

Sequence in context: A350893 A021895 A025160 * A096740 A264595 A026830

Adjacent sequences: A026828 A026829 A026830 * A026832 A026833 A026834

Keyword

nonn

Author

Clark Kimberling

Status

approved