A116155 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).

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, 15, 1, 500, 609, 645, 633, 430, 146, 21, 1, 2485, 2985, 3130, 3142, 2521, 1182, 273, 28, 1, 13262, 15747, 16392, 16561, 14710, 8733, 2849, 470, 36, 1

Offset: 0

Author

Philippe Deléham, Apr 15 2007

Keywords

• nonn
• tabl

Examples

			Triangle begins:
      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,   15,    1;
    500,   609,   645,   633,   430,  146,   21,   1;
   2485,  2985,  3130,  3142,  2521, 1182,  273,  28,  1;
  13262, 15747, 16392, 16561, 14710, 8733, 2849, 470, 36, 1;
		

Links

• G. C. Greubel, Rows n = 0..16 of triangle, flattened

Programs

Mathematica

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 *)

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

Formula

Sum_{k=0..n} T(n,k) = T(n+1,1) = A098742(n+2).

Extensions

Term a(47) corrected in data by G. C. Greubel, May 12 2019