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A094485 T(n, k) = Stirling1(n+1, k) - Stirling1(n, k-1), for 1 <= k <= n. Triangle read by rows. 2

%I

%S -1,2,-2,-6,9,-3,24,-44,24,-4,-120,250,-175,50,-5,720,-1644,1350,-510,

%T 90,-6,-5040,12348,-11368,5145,-1225,147,-7,40320,-104544,105056,

%U -54152,15680,-2576,224,-8,-362880,986256,-1063116,605556,-202041,40824,-4914,324,-9,3628800,-10265760,11727000,-7236800

%N T(n, k) = Stirling1(n+1, k) - Stirling1(n, k-1), for 1 <= k <= n. Triangle read by rows.

%F E.g.f.: -x*y*(1+y)^(x-1). [T(n,k) = n!*[x^k]([y^n] -x*y*(y+1)^(x-1)).]

%F The matrix inverse of the Worpitzky triangle. More precisely:

%F T(n, k) = -n!*InvW(n, k) where InvW is the matrix inverse of A028246. - _Peter Luschny_, May 26 2020

%e Triangle starts:

%e [n\k 1 2 3 4 5 6 7 8]

%e [1] -1;

%e [2] 2, -2;

%e [3] -6, 9, -3;

%e [4] 24, -44, 24, -4;

%e [5] -120, 250, -175, 50, -5;

%e [6] 720, -1644, 1350, -510, 90, -6;

%e [7] -5040, 12348, -11368, 5145, -1225, 147, -7;

%e [8] 40320, -104544, 105056, -54152, 15680, -2576, 224, -8;

%p T := (n, k) -> Stirling1(n+1, k) - Stirling1(n, k-1);

%p seq(seq(T(n, k), k=1..n), n=1..9); # _Peter Luschny_, May 26 2020

%Y Cf. A019538, A028246, A163626.

%Y Cf. A000142, A052881.

%K easy,sign,tabl

%O 1,2

%A _Vladeta Jovovic_, Jun 05 2004

%E Offset of k shifted and edited by _Peter Luschny_, May 26 2020

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Last modified August 11 15:12 EDT 2020. Contains 336428 sequences. (Running on oeis4.)