A094485 T(n, k) = Stirling1(n+1, k) - Stirling1(n, k-1), for 1 <= k <= n. Triangle read by rows.
-1, 2, -2, -6, 9, -3, 24, -44, 24, -4, -120, 250, -175, 50, -5, 720, -1644, 1350, -510, 90, -6, -5040, 12348, -11368, 5145, -1225, 147, -7, 40320, -104544, 105056, -54152, 15680, -2576, 224, -8, -362880, 986256, -1063116, 605556, -202041, 40824, -4914, 324, -9, 3628800, -10265760, 11727000, -7236800
Offset: 1
Examples
Triangle starts: [n\k 1 2 3 4 5 6 7 8] [1] -1; [2] 2, -2; [3] -6, 9, -3; [4] 24, -44, 24, -4; [5] -120, 250, -175, 50, -5; [6] 720, -1644, 1350, -510, 90, -6; [7] -5040, 12348, -11368, 5145, -1225, 147, -7; [8] 40320, -104544, 105056, -54152, 15680, -2576, 224, -8;
Programs
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Maple
T := (n, k) -> Stirling1(n+1, k) - Stirling1(n, k-1); seq(seq(T(n, k), k=1..n), n=1..9); # Peter Luschny, May 26 2020
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Mathematica
Table[StirlingS1[n+1,k]-StirlingS1[n,k-1],{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jul 25 2024 *)
Formula
E.g.f.: -x*y*(1+y)^(x-1). [T(n,k) = n!*[x^k]([y^n] -x*y*(y+1)^(x-1)).]
The matrix inverse of the Worpitzky triangle. More precisely:
T(n, k) = -n!*InvW(n, k) where InvW is the matrix inverse of A028246. - Peter Luschny, May 26 2020
Extensions
Offset of k shifted and edited by Peter Luschny, May 26 2020