A375612 Triangle read by rows: T(n, k) = n! * 4^k * hypergeom([-k], [-n], -1/4).
1, 1, 3, 2, 7, 25, 6, 22, 81, 299, 24, 90, 338, 1271, 4785, 120, 456, 1734, 6598, 25121, 95699, 720, 2760, 10584, 40602, 155810, 598119, 2296777, 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755, 40320, 156240, 605520, 2347080, 9098904, 35278554, 136801778, 530555479, 2057912161
Offset: 0
Examples
Triangle starts: [0] 1; [1] 1, 3; [2] 2, 7, 25; [3] 6, 22, 81, 299; [4] 24, 90, 338, 1271, 4785; [5] 120, 456, 1734, 6598, 25121, 95699; [6] 720, 2760, 10584, 40602, 155810, 598119, 2296777; [7] 5040, 19440, 75000, 289416, 1117062, 4312438, 16651633, 64309755; ...
Crossrefs
Programs
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Mathematica
T[n_, k_] := (-1)^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) = (-1)^k*Sum_{j=0..k} (-4)^(k - j)*binomial(k, k - j)*(n - j)!.