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

%I A094485 #14 Jul 25 2024 12:21:21
%S A094485 -1,2,-2,-6,9,-3,24,-44,24,-4,-120,250,-175,50,-5,720,-1644,1350,-510,
%T A094485 90,-6,-5040,12348,-11368,5145,-1225,147,-7,40320,-104544,105056,
%U A094485 -54152,15680,-2576,224,-8,-362880,986256,-1063116,605556,-202041,40824,-4914,324,-9,3628800,-10265760,11727000,-7236800
%N A094485 T(n, k) = Stirling1(n+1, k) - Stirling1(n, k-1), for 1 <= k <= n. Triangle read by rows.
%F A094485 E.g.f.: -x*y*(1+y)^(x-1). [T(n,k) = n!*[x^k]([y^n] -x*y*(y+1)^(x-1)).]
%F A094485 The matrix inverse of the Worpitzky triangle. More precisely:
%F A094485 T(n, k) = -n!*InvW(n, k) where InvW is the matrix inverse of A028246. - _Peter Luschny_, May 26 2020
%e A094485 Triangle starts:
%e A094485 [n\k    1        2       3      4      5      6     7  8]
%e A094485 [1]    -1;
%e A094485 [2]     2,      -2;
%e A094485 [3]    -6,       9,     -3;
%e A094485 [4]    24,     -44,     24,     -4;
%e A094485 [5]  -120,     250,   -175,     50,    -5;
%e A094485 [6]   720,   -1644,   1350,   -510,    90,    -6;
%e A094485 [7] -5040,   12348, -11368,   5145, -1225,   147,   -7;
%e A094485 [8] 40320, -104544, 105056, -54152, 15680, -2576,  224,  -8;
%p A094485 T := (n, k) -> Stirling1(n+1, k) - Stirling1(n, k-1);
%p A094485 seq(seq(T(n, k), k=1..n), n=1..9); # _Peter Luschny_, May 26 2020
%t A094485 Table[StirlingS1[n+1,k]-StirlingS1[n,k-1],{n,10},{k,n}]//Flatten (* _Harvey P. Dale_, Jul 25 2024 *)
%Y A094485 Cf. A019538, A028246, A163626.
%Y A094485 Cf. A000142, A052881.
%K A094485 easy,sign,tabl
%O A094485 1,2
%A A094485 _Vladeta Jovovic_, Jun 05 2004
%E A094485 Offset of k shifted and edited by _Peter Luschny_, May 26 2020