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Associated Omega numbers of order 3, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.
2

%I #13 Mar 28 2020 05:29:42

%S 1,1,1,1,10,-9,1,28,-504,477,1,55,-4158,78705,-74601,1,91,-18018,

%T 1432431,-27154764,25740261,1,136,-55692,11595870,-923261976,

%U 17503377480,-16591655817,1,190,-139536,60087690,-12529983960,997692516360,-18914487631380,17929265150637

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

%C See the comments in A318254.

%F 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 Triangle starts:

%e [0] 1

%e [1] 1, 1

%e [2] 1, 10, -9

%e [3] 1, 28, -504, 477

%e [4] 1, 55, -4158, 78705, -74601

%e [5] 1, 91, -18018, 1432431, -27154764, 25740261

%e [6] 1, 136, -55692, 11595870, -923261976, 17503377480, -16591655817

%p # The function TNum is defined in A318253.

%p T := (m, n, k) -> `if`(k=0, 1, binomial(m*n-1, m*(n-k))*TNum(m, k)):

%p for n from 0 to 6 do seq(T(3, n, k), k=0..n) od;

%o (Sage) # uses[AssociatedOmegaNumberTriangle from A318254]

%o A318255Triangle = lambda dim: AssociatedOmegaNumberTriangle(3, dim)

%o print(A318255Triangle(8))

%Y T(n, 0) = A060544, T(n, n) = A293951(n+1) (up to signs), row sums are A040000.

%Y Cf. A318146, A318253, A318254 (m=2).

%K sign,tabl

%O 0,5

%A _Peter Luschny_, Aug 26 2018