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A341497 Number of partitions of n with exactly one repeated part and that part is odd. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 5 21:40 EST 2022. Contains 358594 sequences. (Running on oeis4.)