%I #27 May 13 2017 04:58:48

%S 1,0,1,0,1,3,0,1,7,12,0,1,15,50,60,0,1,31,180,390,360,0,1,63,602,2100,

%T 3360,2520,0,1,127,1932,10206,25200,31920,20160,0,1,255,6050,46620,

%U 166824,317520,332640,181440,0,1,511,18660,204630,1020600,2739240,4233600,3780000,1814400,0,1,1023,57002,874500,5921520,21538440,46070640,59875200,46569600,19958400

%N Triangle T(n, k) read by rows: T(n, k) = S2(n, k)*k! + S2(n, k-1)*(k-1)! with the Stirling2 triangle S2 = A048993.

%C This triangle T(n, k) appears in the e.g.f. of the sum of powers SP(n, m) = Sum_{j=0..m} j^n, n >= 0, m >= 0 with 0^0:=1 as ESP(n, t) = exp(t)*(Sum_{k=0..n} T(n, k)*t^k/k! + t^(n+1)/(n+1)), n >= 0.

%C The sub-triangle T(n, k) for 1 <= k <=n, see A028246(n+1,k) (diagonal not needed).

%C For S2(n, m)*m! see A131689.

%C The columns (starting sometimes with n=k) are A000007, A000012, A000225, A028243(n-1), A028244(n-1), A028245(n-1), A032180(n-1), A228909, A228910, A228911, A228912, A228913. See below for the e.g.f.s and o.g.f.s.

%C The row sums are 1 for n=1 and A000629(n) - n! for n >= 1, See A285868.

%F T(n, k) = A131689(n, k) + A131689(n, k-1), 0 <= k <= n, with A131689(n, -1) = 0.

%F T(0, 0) = 1 and T(n, k) = Stirling2(n+1, k)*(k-1)! for n >= k >= 1. For Stirling2 see A048993. Stirling2(n, k)*(k-1)! = A028246(n, k) for n >= k >= 1.

%F Recurrence: T(0, 0) = 1, T(n, n) = (n+1)!/2, T(n, -1) = 0, T(n, k) = 0 if n < k, and T(n, k) = (k-1)*T(n-1, k-1) + k*T(n-1, k), for n > k >= 0.

%F E.g.f. for column k=0 is 1, and for k >= 1: Sum_{j=1..k}((-1)^(k-j) * binomial(k-1, j-1) * exp(j*x)) - x^(k-1).

%F O.g.f. for column k = 0 is 1, and for k >= 1: ((k-1)!*x^(k-1) / Product_{j=1..k} (1-j*x)) - (k-1)!*x^(k-1).

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 ...

%e 0: 1

%e 1: 0 1

%e 2: 0 1 3

%e 3: 0 1 7 12

%e 4: 0 1 15 50 60

%e 5: 0 1 31 180 390 360

%e 6: 0 1 63 602 2100 3360 2520

%e 7: 0 1 127 1932 10206 25200 31920 20160

%e 8: 0 1 255 6050 46620 166824 317520 332640 181440

%e 9: 0 1 511 18660 204630 1020600 2739240 4233600 3780000 1814400

%e ...

%t Table[If[k == 0, Boole[n == 0], StirlingS2[n, k] k! + StirlingS2[n, k - 1] (k - 1)!], {n, 0, 10}, {k, 0, n}] (* _Michael De Vlieger_, May 08 2017 *)

%Y Cf. A000629, A028246, A048993, A131689, A285868.

%K nonn,easy,tabl

%O 0,6

%A _Wolfdieter Lang_, May 03 2017