A240494 Number of partitions of n such that the multiplicity of the greatest part is a part.

0, 1, 0, 1, 3, 4, 6, 8, 13, 18, 27, 36, 51, 67, 92, 120, 162, 208, 276, 352, 457, 579, 743, 931, 1183, 1474, 1851, 2293, 2857, 3515, 4347, 5320, 6532, 7955, 9708, 11762, 14279, 17224, 20798, 24986, 30034, 35935, 43012, 51274, 61125, 72617, 86249, 102120

Offset

0,5

Links

Table of n, a(n) for n=0..47.

Formula

G.f.: Sum_{j>=1} (q^(j^2) + Sum_{k=1..j-1} q^((j+1)*k)) / Product_{k=1..j-1} (1-q^k). - Seiichi Manyama, Mar 15 2026

Example

a(6) counts these 6 partitions:  51, 411, 321, 3111, 2211, 21111.

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; MemberQ[p, Count[p, Mean[p]]]], {n, 0, z}]  (* A240491 *)
Table[Count[f[n], p_ /; MemberQ[p, Count[p, Median[p]]]], {n, 0, z}]  (* A240492 *)
Table[Count[f[n], p_ /; MemberQ[p, Count[p, Min[p]]]], {n, 0, z}]  (* A240493 *)
Table[Count[f[n], p_ /; MemberQ[p, Count[p, Max[p]]]], {n, 0, z}]  (* A240494 *)
Table[Count[f[n], p_ /; MemberQ[p, Count[p, Max[p] - Min[p]]]], {n, 0, z}]  (* A240495 *)

Prog

(PARI) my(N=50, q='q+O('q^N)); concat(0, Vec(sum(j=1, N, (q^j^2+sum(k=1, j-1, q^((j+1)*k)))/prod(k=1, j-1, 1-q^k)))) \\ Seiichi Manyama, Mar 15 2026

Crossrefs

Cf. A006141, A240491 - A240495.

Sequence in context: A367081 A244607 A136483 * A280250 A356081 A004713

Adjacent sequences: A240491 A240492 A240493 * A240495 A240496 A240497

Keyword

nonn,easy

Author

Clark Kimberling, Apr 06 2014

Status

approved