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

0,4

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

Formula

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

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

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

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

Row sums of A330372.

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

Cf. A000700, A066186, A081362, A235324.

Sequence in context: A070916 A078305 A235324 * A072599 A095138 A026475

Adjacent sequences: A330370 A330371 A330372 * A330374 A330375 A330376

Keyword

nonn

Author

Omar E. Pol, Dec 17 2019

Status

approved