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

0,5

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-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

Example

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.

Prog

(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))}

Crossrefs

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

Sequence in context: A302835 A200672 A376581 * A332686 A069999 A271661

Adjacent sequences: A341494 A341495 A341496 * A341498 A341499 A341500

Keyword

nonn,easy

Author

Andrew Howroyd, Feb 13 2021

Status

approved