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%I #9 May 08 2018 06:20:24
%S 199,3753,27267,121367,401565,1089411,2563933,5423365,10557163,
%T 19228309,33165903,54668043,86714993,133092639,198526233,288824425,
%U 411033583,573602401,786556795,1061685087,1412733477,1855611803,2408609589
%N Number of lower triangles of a 5 X 5 -n..n array with all row and column sums zero.
%C Row 5 of A195522.
%H R. H. Hardin, <a href="/A195523/b195523.txt">Table of n, a(n) for n = 1..175</a>
%F Empirical: a(n) = (643/45)*n^6 + (643/15)*n^5 + (2165/36)*n^4 + (293/6)*n^3 + (4423/180)*n^2 + (73/10)*n + 1.
%F Conjectures from _Colin Barker_, May 08 2018: (Start)
%F G.f.: x*(199 + 2360*x + 5175*x^2 + 2346*x^3 + 213*x^4 - 6*x^5 + x^6) / (1 - x)^7.
%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
%F (End)
%e Some solutions for n=4:
%e ..0..........0..........0..........0..........0..........0..........0
%e ..0.0.......-2.2.......-3.3........0.0.......-3.3........3-3........3-3
%e ..0-3.3.....-2.4-2.....-2-1.3......2-2.0......0-1.1.....-3.0.3......2.0-2
%e ..0.3.0-3....2-2.4-4....4-1-4.1....0-1.2-1....1.0-1.0....4-1.1-4...-4-1.4.1
%e ..0.0-3.3.0..2-4-2.4.0..1-1.1-1.0.-2.3-2.1.0..2-2.0.0.0.-4.4-4.4.0.-1.4-2-1.0
%Y Cf. A195522.
%K nonn
%O 1,1
%A _R. H. Hardin_, Sep 20 2011
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