A025150 Number of partitions of n into distinct parts >= 5.

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 20, 23, 26, 30, 35, 39, 45, 51, 58, 66, 75, 84, 96, 108, 122, 137, 155, 173, 195, 219, 245, 274, 307, 342, 383, 427, 475, 529, 589, 654, 727, 807, 894, 991, 1098, 1214, 1343, 1485, 1638, 1809, 1995

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Examples

			a(12) = 2 because we have [12] and [7,5].
		

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=5..70)-1: gser:=series(g,x=0,60): seq(coeff(gser,x,n), n=1..53); # Emeric Deutsch, Apr 17 2006
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`((i-4)*(i+5)/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[#] >= 5 &]; Table[d[n], {n, 16}] (* strict partitions, parts >= 5 *)
Table[Length[d[n]], {n, 40}] (* A025150 for n >= 1 *)
(* Clark Kimberling, Mar 07 2014 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[(i-4)*(i+5)/2Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)
nmax = 100; CoefficientList[Series[Product[1+x^k, {k, 1, nmax}] / ((1+x)*(1+x^2)*(1+x^3)*(1+x^4)), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)

Formula

G.f.: Product_{j>=5} (1+x^j). - Emeric Deutsch, Apr 17 2006

a(n) = A026825(n+4). - R. J. Mathar, Jul 31 2008

a(n) ~ exp(Pi*sqrt(n/3)) / (64*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

G.f.: Sum_{k>=0} x^(k*(k + 9)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 24 2020