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

Author

Andrew Howroyd, Feb 13 2021

Keywords

• nonn

Examples

			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.
		

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.

Crossrefs

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

Programs

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))}

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.