A237825 Number of partitions of n such that 3*(least part) = greatest part.

0, 0, 0, 1, 1, 2, 3, 5, 5, 8, 9, 13, 14, 18, 20, 27, 28, 35, 38, 49, 51, 61, 66, 81, 86, 102, 109, 130, 136, 161, 172, 202, 214, 245, 264, 305, 323, 369, 395, 452, 480, 544, 580, 657, 703, 786, 842, 947, 1008, 1124, 1205, 1340, 1432, 1589, 1702, 1886, 2014

Offset

1,6

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k>=1} x^(4*k)/Product_{j=k..3*k} (1-x^j). - Seiichi Manyama, May 14 2023

Example

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

Mathematica

z = 64; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}]     (* A237825*)
Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}]     (* A237826 *)
Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}]     (* A237827 *)
Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)
Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)
Table[Count[IntegerPartitions[n], _?(3#[[-1]]==#[[1]]&)], {n, 60}] (* Harvey P. Dale, May 14 2023 *)

Prog

(PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=k, 3*k, 1-x^j)))) \\ Seiichi Manyama, May 14 2023

Crossrefs

Cf. A000041, A117086, A237757, A237828, A237829.

Cf. A118096, A237826, A237827.

Sequence in context: A306676 A363058 A341122 * A194939 A135635 A238007

Adjacent sequences: A237822 A237823 A237824 * A237826 A237827 A237828

Keyword

nonn,easy

Author

Clark Kimberling, Feb 16 2014

Status

approved