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%I #18 Dec 29 2023 12:56:09
%S 1,0,1,1,1,1,2,3,3,1,7,9,10,6,1,26,33,36,29,10,1,109,135,145,134,70,
%T 15,1,500,609,645,633,430,146,21,1,2485,2985,3130,3142,2521,1182,273,
%U 28,1,13262,15747,16392,16561,14710,8733,2849,470,36,1
%N Triangle T(n,k) defined by: T(0,0)=1, T(n,k)=0 if k < 0 or k > n, T(n,k) = T(n-1,k-1) + k*T(n-1,k) + Sum_{j>=1} T(n-1,k+j).
%H G. C. Greubel, <a href="/A116155/b116155.txt">Rows n = 0..16 of triangle, flattened</a>
%F Sum_{k=0..n} T(n,k) = T(n+1,1) = A098742(n+2).
%e Triangle begins:
%e 1;
%e 0, 1;
%e 1, 1, 1;
%e 2, 3, 3, 1;
%e 7, 9, 10, 6, 1;
%e 26, 33, 36, 29, 10, 1;
%e 109, 135, 145, 134, 70, 15, 1;
%e 500, 609, 645, 633, 430, 146, 21, 1;
%e 2485, 2985, 3130, 3142, 2521, 1182, 273, 28, 1;
%e 13262, 15747, 16392, 16561, 14710, 8733, 2849, 470, 36, 1;
%t T[0, 0]:= 1; T[n_, k_]:= If[k<0 || k>n, 0, T[n-1, k-1] + k*T[n-1, k] + Sum[T[n-1, k+j], {j, 1, n-k-1}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 10 2019 *)
%o (PARI) {T(n,k) = if(k==0 && n==0, 1, if(k<0 || k>n, 0, T(n-1, k-1) + k*T(n-1, k) + sum(j=1,n-k-1, T(n-1, k+j))))}; \\ _G. C. Greubel_, May 10 2019
%K nonn,tabl
%O 0,7
%A _Philippe Deléham_, Apr 15 2007
%E Term a(47) corrected in data by _G. C. Greubel_, May 12 2019
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