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

1, 1, 1, 1, 10, -9, 1, 28, -504, 477, 1, 55, -4158, 78705, -74601, 1, 91, -18018, 1432431, -27154764, 25740261, 1, 136, -55692, 11595870, -923261976, 17503377480, -16591655817, 1, 190, -139536, 60087690, -12529983960, 997692516360, -18914487631380, 17929265150637

Offset: 0

Author

Peter Luschny, Aug 26 2018

Keywords

• sign
• tabl

Comments

See the comments in A318254.

Examples

			Triangle starts:
[0] 1
[1] 1,   1
[2] 1,  10,     -9
[3] 1,  28,   -504,      477
[4] 1,  55,  -4158,    78705,     -74601
[5] 1,  91, -18018,  1432431,  -27154764,    25740261
[6] 1, 136, -55692, 11595870, -923261976, 17503377480, -16591655817
		

Crossrefs

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

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

Programs

Maple

# The function TNum is defined in A318253.
T := (m, n, k) -> `if`(k=0, 1, binomial(m*n-1, m*(n-k))*TNum(m, k)):
for n from 0 to 6 do seq(T(3, n, k), k=0..n) od;

Sage

# uses[AssociatedOmegaNumberTriangle from A318254]
A318255Triangle = lambda dim: AssociatedOmegaNumberTriangle(3, dim)
print(A318255Triangle(8))

Formula

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.