A330373 Sum of all parts of all self-conjugate partitions of n.

0, 1, 0, 3, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 60, 80, 85, 90, 114, 140, 168, 176, 207, 264, 300, 312, 378, 448, 493, 540, 620, 736, 825, 884, 1015, 1188, 1295, 1406, 1599, 1840, 2009, 2184, 2451, 2772, 3060, 3312, 3666, 4176, 4557, 4900, 5457, 6084, 6625, 7182, 7920, 8792, 9576, 10324, 11328, 12540

Offset: 0

Author

Omar E. Pol, Dec 17 2019

Keywords

• nonn

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.

Examples

			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.
		

Links

• Freddy Barrera, Table of n, a(n) for n = 0..10000

Crossrefs

Row sums of A330372.

For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483, A330370.

Cf. A000700, A066186, A081362, A235324.

Programs

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

Formula

a(n) = n*A000700(n).

a(n) = abs(n*A081362(n)).

a(n) = abs(A235324(n)), n >= 1.