A341496 Number of partitions of n with exactly one repeated part and that part is even.

0, 0, 0, 0, 1, 1, 1, 2, 4, 5, 6, 9, 12, 16, 20, 26, 34, 43, 53, 67, 82, 101, 124, 151, 184, 222, 267, 320, 381, 454, 539, 637, 752, 884, 1038, 1214, 1417, 1651, 1920, 2227, 2578, 2979, 3437, 3957, 4547, 5218, 5980, 6840, 7815, 8914, 10154, 11552, 13122

Offset

0,8

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cristina Ballantine and Mircea Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts, Ramanujan J., 54:1 (2021), 107-112.

Formula

G.f.: (Sum_{k>=1} x^(4*k)/(1 - x^(4*k)) * Product_{k>=1} (1 + x^k).

a(n) = A090867(n) - A341497(n).

a(n) = A341497(n) - A116680(n).

a(n) = A341494(n) for even n; a(n) = A341495(n) for odd n.

Example

The a(4) = 1 partition is: 2+2.
The a(5) = 1 partition is: 1+2+2.
The a(6) = 1 partition is: 2+2+2.
The a(7) = 2 partitions are: 2+2+3, 1+2+2+2.
The a(8) = 4 partitions are: 4+4, 2+2+4, 1+2+2+3, 2+2+2+2.

Prog

(PARI) seq(n)={Vec(sum(k=1, n\4, x^(4*k)/(1 - x^(4*k)) + O(x*x^n)) * prod(k=1, n, 1 + x^k + O(x*x^n)), -(n+1))}

Crossrefs

Cf. A090867, A116680, A341494, A341495, A341497.

Sequence in context: A089969 A166944 A073894 * A056635 A288429 A163116

Adjacent sequences: A341493 A341494 A341495 * A341497 A341498 A341499

Keyword

nonn

Author

Andrew Howroyd, Feb 13 2021

Status

approved