A341494 Number of partitions of n into an even number of parts such that the set of even parts has only one element.

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

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