A025148 Number of partitions of n into distinct parts >= 3.

1, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 15, 17, 20, 24, 28, 32, 38, 44, 51, 59, 68, 78, 91, 103, 118, 136, 155, 176, 201, 228, 259, 294, 332, 375, 425, 478, 538, 607, 681, 764, 858, 961, 1075, 1203, 1343, 1499, 1673, 1863, 2073, 2308, 2564, 2847, 3161, 3504

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.

Crossrefs

Cf. A025147.

Programs

Maple

with(combstruct) ; sys := {L = PowerSet(Sequence(Z,card>2)) }; seq( count([L,sys],size=i), i=0..56 ); # Zerinvary Lajos, Mar 08 2007
A025148 := proc(n) mul(1+x^k,k=3..n+1) ; expand(%) ; coeftayl(%,x=0,n) ; end proc: # R. J. Mathar, Mar 28 2011
# third Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-2)*(i+3)/2 b(n$2):
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 07 2014

Mathematica

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 && Min[#] >= 3 &]; Table[d[n], {n, 16}] (* strict partitions, parts >= 3 *)
Table[Length[d[n]], {n, 40}] (* A025148 for n >= 1 *)
(* Clark Kimberling, Mar 07 2014 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[(i-2)*(i+3)/2Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)

Formula

G.f.: Product_{k>=3} (1+x^k).

a(n) = A096749(n+2). - R. J. Mathar, Jul 31 2008

G.f.: sum(n>=0, x^(n*(n+5)/2) / prod(k=1..n, 1-x^k) ); special case of g.f. for partitions into distinct parts >= L, sum(n>=0, x^(n*(n+2*L-1)/2) / prod(k=1..n, 1-x^k) ). - Joerg Arndt, Mar 24 2011

G.f.: sum(n>=2, x^(n*(n+1)/2-3) / prod(k=1..n-2, 1-x^k) ), a special case of the g.f. for partitions into distinct parts >= L, sum(n>=L-1, x^(n*(n+1)/2-L*(L-1)/2) / prod(k=1..n-(L-1), 1-x^k) ). - Joerg Arndt, Mar 27 2011

a(n) + a(n+1) + a(n+2) + a(n+3) = A000009(n+3). - Vaclav Kotesovec, Oct 22 2015

a(n) ~ 1/4 * A000009(n). - Vaclav Kotesovec, Oct 22 2015