A375613 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], 1/4).
1, 1, 5, 2, 9, 41, 6, 26, 113, 493, 24, 102, 434, 1849, 7889, 120, 504, 2118, 8906, 37473, 157781, 720, 3000, 12504, 52134, 217442, 907241, 3786745, 5040, 20880, 86520, 358584, 1486470, 6163322, 25560529, 106028861, 40320, 166320, 686160, 2831160, 11683224, 48219366, 199040786, 821723673, 3392923553
Offset: 0
Examples
Triangle starts: [0] 1; [1] 1, 5; [2] 2, 9, 41; [3] 6, 26, 113, 493; [4] 24, 102, 434, 1849, 7889; [5] 120, 504, 2118, 8906, 37473, 157781; [6] 720, 3000, 12504, 52134, 217442, 907241, 3786745; [7] 5040, 20880, 86520, 358584, 1486470, 6163322, 25560529, 106028861; ...
Crossrefs
Programs
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Mathematica
T[n_, k_] := Sum[4^(k - j)*Binomial[k, k - j]*(n - j)!, {j, 0, k}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
Formula
T(n, k) = Sum_{j=0..k} 4^(k - j)*binomial(k, k - j)*(n - j)!.