A291980 Triangle read by rows, T(n, k) = n!*[t^k] ([x^n] exp(x*t)/(1 - log(1+x))) for 0<=k<=n.
1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 4, 8, 6, 4, 1, 14, 20, 20, 10, 5, 1, 38, 84, 60, 40, 15, 6, 1, 216, 266, 294, 140, 70, 21, 7, 1, 600, 1728, 1064, 784, 280, 112, 28, 8, 1, 6240, 5400, 7776, 3192, 1764, 504, 168, 36, 9, 1
Offset: 0
Examples
Triangle starts: [1] [1, 1] [1, 2, 1] [2, 3, 3, 1] [4, 8, 6, 4, 1] [14, 20, 20, 10, 5, 1] [38, 84, 60, 40, 15, 6, 1] [216, 266, 294, 140, 70, 21, 7, 1] [600, 1728, 1064, 784, 280, 112, 28, 8, 1]
Crossrefs
Programs
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Maple
T_row := proc(n) exp(x*t)/(1 - log(1+x)): series(%, x, n+1): seq(n!*coeff(coeff(%,x,n), t, k), k=0..n) end: seq(T_row(n), n=0..10);
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Mathematica
T[n_, k_] := Binomial[n, n - k]*Sum[j!*StirlingS1[n - k, j], {j, 0, n - k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, May 12 2024 *)
Formula
T(n, k) = binomial(n, n - k)*Sum_{j=0..n - k} j!*Stirling1(n - k, j). - Detlef Meya, May 12 2024
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