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A341494 Number of partitions of n into an even number of parts such that the set of even parts has only one element. 5
0, 0, 0, 1, 1, 3, 1, 7, 4, 13, 6, 23, 12, 39, 20, 63, 34, 98, 53, 150, 82, 225, 124, 329, 184, 475, 267, 676, 381, 948, 539, 1317, 752, 1810, 1038, 2460, 1417, 3319, 1920, 4442, 2578, 5897, 3437, 7780, 4547, 10200, 5980, 13285, 7815, 17214, 10154, 22191, 13122 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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.: (P(x,1) + P(x,-1))/2 where P(x,c) = (Sum_{k>=1} c*x^(2*k)/(1-c*x^(2*k))) / (Product_{k>=1} 1-c*x^(2*k-1)).

a(n) = A090867(n) - A341495(n).

EXAMPLE

The a(3) = 1 partition is: 1+2.

The a(4) = 1 partition is: 2+2.

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

MATHEMATICA

P[n_, c_] := c*Sum[x^(2k)/(1 - c*x^(2k)) + O[x]^n, {k, 1, n/2}]/

Product[1 - c*x^(2k - 1) + O[x]^n, {k, 1, n/2}];

CoefficientList[(P[100, 1] + P[100, -1])/2, x] (* Jean-François Alcover, May 24 2021, from PARI code *)

PROG

(PARI)

P(n, c)={c*sum(k=1, n\2, x^(2*k)/(1-c*x^(2*k)) + O(x*x^n))/prod(k=1, n\2, 1-c*x^(2*k-1) + O(x*x^n))}

seq(n)={Vec(P(n, 1) + P(n, -1), -(n+1))/2}

CROSSREFS

Cf. A090867, A341495, A341496, A341497.

Sequence in context: A010603 A269423 A328461 * A210198 A271258 A100584

Adjacent sequences: A341491 A341492 A341493 * A341495 A341496 A341497

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Feb 13 2021

STATUS

approved

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Last modified January 29 23:01 EST 2023. Contains 359939 sequences. (Running on oeis4.)