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A255001 Number of partitions of 4n into distinct parts with equal sums of odd and even parts.

%I A255001 #17 Dec 11 2020 04:03:53
%S A255001 1,0,1,2,4,6,12,15,30,40,70,96,165,216,352,486,736,988,1518,1998,2944,
%T A255001 3952,5607,7488,10614,13916,19305,25536,34854,45568,61864,80240,
%U A255001 107640,139776,184832,238680,314628,402800,526176,673652,872592,1110060,1431704
%N A255001 Number of partitions of 4n into distinct parts with equal sums of odd and even parts.
%H A255001 Alois P. Heinz, <a href="/A255001/b255001.txt">Table of n, a(n) for n = 0..1000</a>
%F A255001 a(n) = A000009(n) * A069910(n) = A000009(n) * A000700(2n).
%F A255001 a(n) ~ exp(2*Pi*sqrt(n/3)) / (16*sqrt(6)*n^(3/2)). - _Vaclav Kotesovec_, Dec 11 2020
%e A255001 a(0) = 1: [], the empty partition.
%e A255001 a(1) = 0.
%e A255001 a(2) = 1: [4,3,1].
%e A255001 a(3) = 2: [6,5,1], [5,4,2,1].
%e A255001 a(4) = 4: [8,7,1], [8,5,3], [7,6,2,1], [6,5,3,2].
%p A255001 g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A255001      `if`(n=0, 1, g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
%p A255001     end:
%p A255001 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A255001      `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i-2))))
%p A255001     end:
%p A255001 a:= n-> g(n$2)*b(2*n, 2*n-1):
%p A255001 seq(a(n), n=0..50);
%t A255001 g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, 1, g[n, i - 1] + If[i > n, 0, g[n - i, i - 1]]]];
%t A255001 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 - 2]]]];
%t A255001 a[n_] := g[n, n] b[2n, 2n-1];
%t A255001 a /@ Range[0, 50] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)
%Y A255001 Cf. A000009, A000700, A069910, A239241, A249914.
%K A255001 nonn
%O A255001 0,4
%A A255001 _Alois P. Heinz_, Feb 11 2015