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A195522 T(n,k) = Number of lower triangles of an n X n -k..k array with all row and column sums zero.

%I A195522 #8 May 11 2022 03:27:28
%S A195522 1,1,1,1,1,3,1,1,5,15,1,1,7,65,199,1,1,9,175,3753,6247,1,1,11,369,
%T A195522 27267,860017,505623,1,1,13,671,121367,23663523,839301197,105997283,1,
%U A195522 1,15,1105,401565,286168923,122092290831,3535646416019,58923059879,1,1,17
%N A195522 T(n,k) = Number of lower triangles of an n X n -k..k array with all row and column sums zero.
%C A195522 Table starts
%C A195522 ....1......1........1.........1..........1...........1...........1.......1....1
%C A195522 ....1......1........1.........1..........1...........1...........1.......1....1
%C A195522 ....3......5........7.........9.........11..........13..........15......17...19
%C A195522 ...15.....65......175.......369........671........1105........1695....2465.3439
%C A195522 ..199...3753....27267....121367.....401565.....1089411.....2563933.5423365
%C A195522 .6247.860017.23663523.286168923.2106810049.11131321791.46387885537
%H A195522 R. H. Hardin, <a href="/A195522/b195522.txt">Table of n, a(n) for n = 1..75</a>
%F A195522 Empirical for rows:
%F A195522 T(2,k) = 1
%F A195522 T(3,k) = 2*k + 1
%F A195522 T(4,k) = 4*k^3 + 6*k^2 + 4*k + 1
%F A195522 T(5,k) = (643/45)*k^6 + (643/15)*k^5 + (2165/36)*k^4 + (293/6)*k^3 + (4423/180)*k^2 + (73/10)*k + 1
%F A195522 T(6,k) = (7389349/90720)*k^10 + (7389349/18144)*k^9 + (836251/864)*k^8 + (4318165/3024)*k^7 + (6254923/4320)*k^6 + (4563293/4320)*k^5 + (10247161/18144)*k^4 + (249983/1134)*k^3 + (21959/360)*k^2 + (3469/315)*k + 1
%e A195522 Some solutions for n=5 k=6
%e A195522 ..0..........0..........0..........0..........0..........0..........0
%e A195522 ..0.0.......-2.2........6-6.......-1.1........5-5.......-4.4.......-4.4
%e A195522 .-1.3-2.....-6.0.6.....-6.6.0.....-1.5-4.....-6.4.2......3-6.3.....-4.1.3
%e A195522 ..6-3-2-1....4-4-4.4....5.3-5-3....0-5.3.2....0.4-3-1...-5.5.1-1....5-2-4.1
%e A195522 .-5.0.4.1.0..4.2-2-4.0.-5-3.5.3.0..2-1.1-2.0..1-3.1.1.0..6-3-4.1.0..3-3.1-1.0
%Y A195522 Row 4 is A005917(n+1).
%K A195522 nonn,tabl
%O A195522 1,6
%A A195522 _R. H. Hardin_, Sep 20 2011