A026830 Number of partitions of n into distinct parts, the least being 9.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 9, 9, 11, 12, 14, 15, 18, 19, 22, 24, 27, 29, 33, 36, 40, 44, 49, 54, 60, 66, 73, 81, 89, 98, 108, 119, 130, 144, 157, 173, 189, 208, 227, 250, 272, 299, 326, 358, 389

Offset

0,31

Links

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

Formula

a(n) = A025155(n-9), n>9. - R. J. Mathar, Jul 31 2008

G.f.: x^9*Product_{j>=10} (1+x^j). - R. J. Mathar, Jul 31 2008

G.f.: Sum_{k>=1} x^(k*(k + 17)/2) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

Maple

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-9)*(i+10)/2<n, 0,
       add(b(n-i*j, i-1), j=0..min(1, n/i))))
    end:
a:= n-> `if`(n<9, 0, b(n-9$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 07 2014

Mathematica

dp9[n_]:=Module[{ips=IntegerPartitions[n]}, Length[Select[ips, Min[#] == 9 && Length[#]==Length[Union[#]]&]]]; Table[dp9[n], {n, 0, 80}] (* Harvey P. Dale, Oct 23 2015 *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 9], {n, 66}]] (* Robert Price, Jun 13 2020 *)

Crossrefs

Cf. A025147, A025155.

Sequence in context: A026831 A096740 A264595 * A025155 A286744 A111684

Adjacent sequences: A026827 A026828 A026829 * A026831 A026832 A026833

Keyword

nonn

Author

Clark Kimberling

Status

approved