OFFSET
0,4
COMMENTS
a(n) is the sum of all parts of all partitions of n whose Ferrers diagrams are symmetric.
The k-th part of a partition equals the number of parts >= k of its conjugate partition. Hence, the k-th part of a self-conjugate partition equals the number of parts >= k.
The k-th rank of a partition is the k-th part minus the number of parts >= k. Thus all ranks of a conjugate-partitions are zero. Therefore, a(n) is also the sum of all parts of all partitions of n whose n ranks are zero, n >= 1. For more information about the k-th ranks see A208478.
LINKS
Freddy Barrera, Table of n, a(n) for n = 0..10000
EXAMPLE
For n = 10 there are only two partitions of 10 whose Ferrers diagram are symmetric, they are [5, 2, 1, 1, 1] and [4, 3, 2, 1] as shown below:
* * * * *
* *
*
*
*
* * * *
* * *
* *
*
The sum of all parts of these partitions is 5 + 2 + 1 + 1 + 1 + 4 + 3 + 2 + 1 = 20, so a(10) = 20.
Also, in accordance with the first formula; a(10) = 2*10 = 20.
PROG
(PARI) seq(n)={Vec(deriv(exp(sum(k=1, n, x^k/(k*(1 - (-x)^k)) + O(x*x^n)))), -(n+1))} \\ Andrew Howroyd, Dec 31 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Dec 17 2019
STATUS
approved