

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
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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), 107112.


FORMULA

G.f.: (Sum_{k>=1} x^(4*k2)/(1  x^(4*k2)) * 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*k2)/(1  x^(4*k2)) + 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



