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%I #33 Dec 11 2020 04:03:42
%S 1,1,4,12,30,70,165,330,704,1380,2688,4984,9394,16665,29970,52096,
%T 90090,152064,257180,423360,697851,1129392,1819632,2891520,4583250,
%U 7162364,11161752,17211180,26427544,40208520,60971520,91641748,137290956,204198876,302530560
%N Number of partitions of 4n with equal sums of odd and even parts.
%H Alois P. Heinz, <a href="/A249914/b249914.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A000041(n) * A035294(n) = A000041(n) * A000009(2n).
%F a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (16*6^(3/4)*n^(7/4)). - _Vaclav Kotesovec_, Dec 11 2020
%e a(0) = 1: [], the empty partition.
%e a(1) = 1: [2,1,1].
%e a(2) = 4: [4,3,1], [4,1,1,1,1], [3,2,2,1], [2,2,1,1,1,1].
%e a(3) = 12: [6,5,1], [6,3,3], [6,3,1,1,1], [6,1,1,1,1,1,1], [5,4,2,1], [5,2,2,2,1], [4,3,3,2], [4,3,2,1,1,1], [4,2,1,1,1,1,1,1], [3,3,2,2,2], [3,2,2,2,1,1,1], [2,2,2,1,1,1,1,1,1].
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
%p end:
%p a:= n-> combinat[numbpart](n) *b(2*n, 2*n-1):
%p seq(a(n), n=0..50);
%t b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]]];
%t a[n_] := PartitionsP[n] b[2n, 2n-1];
%t a /@ Range[0, 50] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)
%Y Cf. A000009, A000041, A035294, A045931, A255001.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Feb 11 2015
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