A238590 Number of partitions p of n such that 3*min(p) is a part of p.

0, 0, 0, 1, 1, 2, 3, 6, 7, 12, 16, 25, 32, 46, 61, 86, 110, 149, 192, 257, 326, 425, 538, 694, 871, 1107, 1381, 1740, 2154, 2689, 3313, 4103, 5024, 6176, 7529, 9201, 11157, 13554, 16365, 19784, 23782, 28610, 34260, 41039, 48958, 58405, 69431, 82525, 97775

Offset

1,6

Links

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

Formula

G.f.: Sum_{k>=1} x^(4*k)/Product_{j>=k} (1-x^j). - Seiichi Manyama, May 17 2023

Example

a(7) = 3 counts these partitions:  331, 3211, 31111.

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-4*i, i), i=1..n/4):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 03 2014

Mathematica

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 3*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-4i, i], {i, 1, n/4}];
Array[a, 60] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)

Prog

(PARI) my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=k, N, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

Crossrefs

Cf. A117989, A238589, A238591.

Cf. A237825, A363066.

Sequence in context: A365411 A064689 A240175 * A144120 A375401 A073712

Adjacent sequences: A238587 A238588 A238589 * A238591 A238592 A238593

Keyword

nonn,easy

Author

Clark Kimberling, Mar 01 2014

Status

approved