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.

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. A090867, A116680, A341494, A341495, A341496.

Sequence in context: A354531 A302835 A200672 * A332686 A069999 A271661

Adjacent sequences: A341494 A341495 A341496 * A341498 A341499 A341500

Keyword

nonn

Author

Andrew Howroyd, Feb 13 2021

Status

approved