A025149 Number of partitions of n into distinct parts >= 4.

1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 17, 20, 23, 27, 31, 36, 41, 47, 55, 62, 71, 81, 93, 105, 120, 135, 154, 174, 197, 221, 251, 281, 317, 356, 400, 447, 502, 561, 628, 701, 782, 871, 972, 1081, 1202, 1336, 1483, 1645, 1825, 2021, 2237, 2476, 2736

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

g:=product(1+x^(j), j=4..54): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=0..53); # Zerinvary Lajos, Mar 09 2007
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-3)*(i+4)/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[#] >= 4 &]; Table[d[n], {n, 12}] (* strict partitions, parts >= 4 *)
Table[Length[d[n]], {n, 40}] (* A025149 for n >= 1 *)
(* Clark Kimberling, Mar 07 2014 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[(i-3)*(i+4)/2Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)

Formula

a(n) = A026824(n+3). - R. J. Mathar, Jul 31 2008

G.f.: Product_{j>=4} (1+x^j). - R. J. Mathar, Jul 31 2008

G.f.: sum(n>=0, x^(n*(n+7)/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

From Vaclav Kotesovec, Oct 22 2015: (Start)

a(n) + a(n+1) + a(n+2) + 2*a(n+3) + a(n+4) + a(n+5) + a(n+6) = A000009(n+6).

a(n) ~ 1/8 * A000009(n).

(End)

Extensions

More terms from N. J. A. Sloane, Sep 28 2008