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A318255 Associated Omega numbers of order 3, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.

%I A318255 #13 Mar 28 2020 05:29:42
%S A318255 1,1,1,1,10,-9,1,28,-504,477,1,55,-4158,78705,-74601,1,91,-18018,
%T A318255 1432431,-27154764,25740261,1,136,-55692,11595870,-923261976,
%U A318255 17503377480,-16591655817,1,190,-139536,60087690,-12529983960,997692516360,-18914487631380,17929265150637
%N A318255 Associated Omega numbers of order 3, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.
%C A318255 See the comments in A318254.
%F A318255 T(m, n, k) = binomial(m*n-1, m*(n-k))*A318253(m, k) for k>0 and 1 for k=0. We consider here the case m=3.
%e A318255 Triangle starts:
%e A318255 [0] 1
%e A318255 [1] 1,   1
%e A318255 [2] 1,  10,     -9
%e A318255 [3] 1,  28,   -504,      477
%e A318255 [4] 1,  55,  -4158,    78705,     -74601
%e A318255 [5] 1,  91, -18018,  1432431,  -27154764,    25740261
%e A318255 [6] 1, 136, -55692, 11595870, -923261976, 17503377480, -16591655817
%p A318255 # The function TNum is defined in A318253.
%p A318255 T := (m, n, k) -> `if`(k=0, 1, binomial(m*n-1, m*(n-k))*TNum(m, k)):
%p A318255 for n from 0 to 6 do seq(T(3, n, k), k=0..n) od;
%o A318255 (Sage) # uses[AssociatedOmegaNumberTriangle from A318254]
%o A318255 A318255Triangle = lambda dim: AssociatedOmegaNumberTriangle(3, dim)
%o A318255 print(A318255Triangle(8))
%Y A318255 T(n, 0) = A060544, T(n, n) = A293951(n+1) (up to signs), row sums are A040000.
%Y A318255 Cf. A318146, A318253, A318254 (m=2).
%K A318255 sign,tabl
%O A318255 0,5
%A A318255 _Peter Luschny_, Aug 26 2018