A237756 Number of partitions of n such that 3*(greatest part) = (number of parts).

0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 6, 7, 10, 10, 13, 14, 19, 21, 27, 31, 40, 45, 55, 64, 79, 91, 111, 127, 154, 177, 211, 243, 290, 333, 394, 455, 538, 618, 726, 834, 977, 1121, 1304, 1495, 1738, 1989, 2302, 2633, 3041, 3473, 3999, 4562, 5241

Offset

1,11

Comments

Also, the number of partitions of n such that (greatest part) = 3*(number of parts).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{k>=1} x^(4*k-1) * Product_{j=1..k-1} (1-x^(3*k+j-1)/(1-x^j). - Seiichi Manyama, Jan 24 2022

Example

a(15) = 4 counts these partitions: [12,1,1,1], [9,5,1], [9,4,2], [9,3,3].

Mathematica

z = 50; Table[Count[IntegerPartitions[n], p_ /; Max[p] = = 3 Length[p]], {n, z}]

Prog

(PARI) my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(4*k-1)*prod(j=1, k-1, (1-x^(3*k+j-1))/(1-x^j))))) \\ Seiichi Manyama, Jan 24 2022

Crossrefs

Column 3 of A350879.

Cf. A064173, A237753, A350892, A350894.

Sequence in context: A326115 A355746 A239485 * A010334 A010578 A328583

Adjacent sequences: A237753 A237754 A237755 * A237757 A237758 A237759

Keyword

nonn,easy

Author

Clark Kimberling, Feb 13 2014

Status

approved