A025155 Number of partitions of n into distinct parts >= 10.

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, 427, 464, 508, 553

Offset

0,22

Links

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

Formula

a(n) = A026830(n+9). - R. J. Mathar, Jul 31 2008

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

G.f.: Sum_{k>=0} x^(k*(k + 19)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 24 2020

Example

Say n = 11. Then there is one and only one partition of n into distinct parts each of which is >= 10, namely 11 = 11. Hence a(11) = 1.

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-> b(n$2):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 07 2014

Mathematica

dpgt10[n_]:=Module[{prtns=Select[IntegerPartitions[n], Min[#]>9&]}, Count[ prtns, _?(Max[Transpose[Tally[#]][[2]]]==1&)]]; Join[{1}, Array[ dpgt10, 80]] (* Harvey P. Dale, Jun 04 2012 *)
(* also *)
d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 10 &]; Table[d[n], {n, 25}] (* str partitions, parts >= 10 *) Table[Length[d[n]], {n, 40}] (* A025155 for n >= 1 *)
(* Clark Kimberling, Mar 07 2014 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[(i-9)*(i+10)/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, Nov 17 2015, after Alois P. Heinz *)

Crossrefs

Cf. A026831, A025147.

Sequence in context: A096740 A264595 A026830 * A286744 A111684 A025159

Adjacent sequences: A025152 A025153 A025154 * A025156 A025157 A025158

Keyword

nonn

Author

Clark Kimberling

Status

approved