A341497 Number of partitions of n with exactly one repeated part and that part is odd.

0, 0, 1, 1, 2, 3, 5, 7, 9, 13, 17, 23, 30, 39, 49, 63, 78, 98, 122, 150, 184, 225, 272, 329, 397, 475, 567, 676, 802, 948, 1121, 1317, 1545, 1810, 2112, 2460, 2863, 3319, 3842, 4442, 5123, 5897, 6782, 7780, 8913, 10200, 11648, 13285, 15136, 17214, 19555, 22191, 25143

Offset: 0

Author

Andrew Howroyd, Feb 13 2021

Keywords

• nonn
• easy

Examples

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

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. A067588, A090867, A116676, A116680, A341494, A341495, A341496.

Programs

PARI

seq(n)={Vec(sum(k=1, (n+2)\4, x^(4*k-2)/(1 - x^(4*k-2)) + 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-2)/(1 - x^(4*k-2))) * Product_{k>=1} (1 + x^k).

a(n) = A090867(n) - A341496(n).

a(n) = A116680(n) + A341496(n).

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

a(n) = (A067588(n) - A116676(n))/2. - Peter Bala, Jan 13 2025