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

%I A330373 #32 Jan 03 2020 16:20:53
%S A330373 0,1,0,3,4,5,6,7,16,18,20,22,36,39,42,60,80,85,90,114,140,168,176,207,
%T A330373 264,300,312,378,448,493,540,620,736,825,884,1015,1188,1295,1406,1599,
%U A330373 1840,2009,2184,2451,2772,3060,3312,3666,4176,4557,4900,5457,6084,6625,7182,7920,8792,9576,10324,11328,12540
%N A330373 Sum of all parts of all self-conjugate partitions of n.
%C A330373 a(n) is the sum of all parts of all partitions of n whose Ferrers diagrams are symmetric.
%C A330373 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.
%C A330373 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.
%H A330373 Freddy Barrera, <a href="/A330373/b330373.txt">Table of n, a(n) for n = 0..10000</a>
%F A330373 a(n) = n*A000700(n).
%F A330373 a(n) = abs(n*A081362(n)).
%F A330373 a(n) = abs(A235324(n)), n >= 1.
%e A330373 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:
%e A330373   * * * * *
%e A330373   * *
%e A330373   *
%e A330373   *
%e A330373   *
%e A330373             * * * *
%e A330373             * * *
%e A330373             * *
%e A330373             *
%e A330373 The sum of all parts of these partitions is 5 + 2 + 1 + 1 + 1 + 4 + 3 + 2 + 1 = 20, so a(10) = 20.
%e A330373 Also, in accordance with the first formula; a(10) = 2*10 = 20.
%o A330373 (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
%Y A330373 Row sums of A330372.
%Y A330373 For "k-th rank" of a partition see also: A181187, A208478, A208479, A208482, A208483, A330370.
%Y A330373 Cf. A000700, A066186, A081362, A235324.
%K A330373 nonn
%O A330373 0,4
%A A330373 _Omar E. Pol_, Dec 17 2019