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
EXAMPLE
The a(2) = 1 partition is: 2.
The a(4) = 2 partitions are: 4, 1+1+2.
The a(5) = 1 partition is: 1+2+2.
The a(6) = 5 partitions are: 6, 1+1+4, 1+2+3, 2+2+2, 1+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
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Feb 13 2021
STATUS
approved