A238591 Number of partitions p of n such that 4*min(p) is a part of p.

0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 16, 23, 32, 45, 60, 81, 109, 144, 190, 247, 320, 412, 529, 675, 854, 1078, 1355, 1695, 2117, 2626, 3251, 4010, 4932, 6047, 7394, 9012, 10959, 13290, 16083, 19407, 23379, 28090, 33689, 40317, 48158, 57406, 68324, 81155, 96248

Offset: 1

Author

Clark Kimberling, Mar 01 2014

Keywords

• nonn
• easy

Examples

			a(9) = 5 counts these partitions:  441, 4311, 4221, 42111, 411111.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Crossrefs

Cf. A117989, A238589, A238590.

Cf. A237826, A363067.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= n-> add(b(n-5*i, i), i=1..n/5):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 03 2014

Mathematica

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 4*Min[p]]], {n, 50}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := Sum[b[n - 5*i, i], {i, 1, n/5}];
Array[a, 60] (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

PARI

my(N=50, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=k, N, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

Formula

G.f.: Sum_{k>=1} x^(5*k)/Product_{j>=k} (1-x^j). - Seiichi Manyama, May 17 2023

From Vaclav Kotesovec, Jun 19 2025: (Start)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 97*Pi/(24*sqrt(6))) / sqrt(n)).

A000041(n) - a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3*sqrt(2)*n^(3/2)). (End)