%I #11 Dec 15 2017 13:45:45

%S 1,1,2,2,4,8,8,14,26,50,50,82,138,242,442,442,692,1102,1792,3002,5212,

%T 5212,7864,12016,18628,29380,47392,78664,78664,115148,170196,254308,

%U 384704,590364,922108,1472756,1472756,2102068,3023252,4384820,6419284

%N Triangle, read by rows, where T(n,k) = T(n,k-1) + n*T(n-1,k-1) for n>0 and k>0, with T(n,0) = T(n-1,n-1) for n>0 and T(0,0) = 1.

%C Column 0 and main diagonal (offset) equals A110083.

%H G. C. Greubel, <a href="/A132007/b132007.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F T(n+1,0) = Sum_{k=0..n} (n!/k!)*C(n,k)*T(k,0) for n>=0 with T(0,0)=1.

%e Triangle begins:

%e 1;

%e 1, 2;

%e 2, 4, 8;

%e 8, 14, 26, 50;

%e 50, 82, 138, 242, 442;

%e 442, 692, 1102, 1792, 3002, 5212;

%e 5212, 7864, 12016, 18628, 29380, 47392, 78664;

%e 78664, 115148, 170196, 254308, 384704, 590364, 922108, 1472756;

%e 1472756, 2102068, 3023252, 4384820, 6419284, 9496916, 14219828, 21596692, 33378740; ...

%t T[n_, k_] := T[n, k] = If[k < 0 || n < k, 0, If[n == 0 && k == 0, 1, If[k == 0, T[n - 1, n - 1], T[n, k - 1] + n*T[n - 1, k - 1]]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Dec 15 2017 *)

%o (PARI) T(n,k)=if(k<0 || n<k,0,if(n==0 && k==0,1,if(k==0,T(n-1,n-1),T(n,k-1)+n*T(n-1,k-1))))

%Y Cf. A110083 (column 0); A132008 (row sums).

%K nonn,tabl

%O 0,3

%A _Paul D. Hanna_, Aug 07 2007