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A203292 Number of arrays of 4 nondecreasing integers in -n..n with sum zero and equal numbers greater than zero and less than zero.

%I A203292 #8 Jun 04 2018 04:29:16
%S A203292 3,6,12,21,35,54,80,113,155,206,268,341,427,526,640,769,915,1078,1260,
%T A203292 1461,1683,1926,2192,2481,2795,3134,3500,3893,4315,4766,5248,5761,
%U A203292 6307,6886,7500,8149,8835,9558,10320,11121,11963,12846,13772,14741,15755
%N A203292 Number of arrays of 4 nondecreasing integers in -n..n with sum zero and equal numbers greater than zero and less than zero.
%C A203292 Row 2 of A203291.
%H A203292 R. H. Hardin, <a href="/A203292/b203292.txt">Table of n, a(n) for n = 1..210</a>
%F A203292 Empirical: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5).
%F A203292 Conjectures from _Colin Barker_, Jun 04 2018: (Start)
%F A203292 G.f.: x*(3 - 3*x + 3*x^3 - x^4) / ((1 - x)^4*(1 + x)).
%F A203292 a(n) = (24 + 32*n + 6*n^2 + 4*n^3)/24 for n even.
%F A203292 a(n) = (30 + 32*n + 6*n^2 + 4*n^3)/24 for n odd.
%F A203292 (End)
%e A203292 All solutions for n=3:
%e A203292 .-2...-3....0...-2...-2...-1...-3...-3...-3...-3...-1...-2
%e A203292 ..0....0....0...-1...-2....0...-1...-2...-1...-3...-1...-2
%e A203292 ..0....0....0....1....2....0....2....2....1....3....1....1
%e A203292 ..2....3....0....2....2....1....2....3....3....3....1....3
%Y A203292 Cf. A203291.
%K A203292 nonn
%O A203292 1,1
%A A203292 _R. H. Hardin_, Dec 31 2011