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

1,7

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A117989, A238589, A238590.

Cf. A237826, A363067.

Sequence in context: A173599 A006304 A344650 * A039847 A046938 A060677

Adjacent sequences: A238588 A238589 A238590 * A238592 A238593 A238594

Keyword

nonn,easy

Author

Clark Kimberling, Mar 01 2014

Status

approved