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%I #24 Nov 22 2022 22:23:26
%S 1,-3,2,12,-24,6,-60,280,-180,24,360,-3600,4500,-1440,120,-2520,52080,
%T -113400,65520,-12600,720,20160,-846720,3034080,-2822400,940800,
%U -120960,5040,-181440,15361920,-87635520,123451776,-63504000,13789440,-1270080,40320
%N Triangle of numbers T(n,k) = (-1)^(n-k)*(n+1)!*Stirling2(n,k)/(k+1)
%C Coefficients of Sum_{k=1..n} T(n,k)*(m+k)!/((m-1)!*(n+1)!) = Sum_{i=1..m} i^n.
%F T(n,k) = (-1)^(n-k)*(n+1)!/(k+1)! * Sum_{i=0..k} (-1)^i*binomial(k,i)*(k-i)^n.
%F T(n+1,k+1) = (-1)^(n+k)*(n+2)*(k+1)*(abs(T(n,k)/(k+2)) + abs(T(n,k+1))) with T(n,1) = (-1)^(n+1)*(n+1)!/2 and T(n,k) = 0 if n < k.
%e The triangle T(n,k) begins:
%e n\k 1 2 3 4 5 6 7 8
%e 1: 1
%e 2: -3 2
%e 3: 12 -24 6
%e 4: -60 280 -180 24
%e 5: 360 -3600 4500 -1440 120
%e 6: -2520 52080 -113400 65520 -12600 720
%e 7: 20160 -846720 3034080 -2822400 940800 -120960 5040
%e 8: -181440 15361920 -87635520 123451776 -63504000 13789440 -1270080 40320
%e ---------------------------------------------------------------------------------
%t T[n_, k_] := (-1)^(n - k) * (n + 1)!/(k + 1) * StirlingS2[n, k]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Sep 01 2022 *)
%o (Python)
%o def A(d):
%o A = [[0 for col in range(d)] for row in range(d)]
%o for i in range(d):
%o A[0][i] = 1
%o for i in range(1, d):
%o for j in range(i, d):
%o A[i][j] = (A[i][j - 1] + A[i - 1][j - 1])*(i + 1)
%o for i in range(d):
%o for j in range(i, d):
%o for k in range(i+3, j+3):
%o A[i][j] *= k
%o a = []
%o for i in range(d):
%o for j in range(i+1):
%o a.append((-1)**(i+j)*A[j][i])
%o return(a)
%o (PARI) T(n,k) = (-1)^(n-k)*(n+1)!*stirling(n,k,2)/(k+1); \\ _Michel Marcus_, Sep 26 2022
%Y Closely related to A019538.
%K sign,tabl
%O 1,2
%A _Samuel Gantner_, Aug 31 2022