A258220 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258219(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 4, 6, 1, 25, 49, 15, 1, 208, 498, 217, 28, 1, 2146, 6016, 3360, 635, 45, 1, 26368, 84042, 56728, 13997, 1475, 66, 1, 375733, 1332661, 1046619, 316281, 43974, 2954, 91, 1, 6092032, 23660034, 21053089, 7479444, 1283817, 114576, 5334, 120, 1

Offset: 0

Author

Alois P. Heinz, May 23 2015

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
:     1;
:     1,     1;
:     4,     6,     1;
:    25,    49,    15,     1;
:   208,   498,   217,    28,    1;
:  2146,  6016,  3360,   635,   45,  1;
: 26368, 84042, 56728, 13997, 1475, 66, 1;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Feynman diagram

Crossrefs

Column k=0 gives A005411 (for n>0).

Main diagonal and lower diagonal give: A000012, A000384(n+1).

Row sums give A258221.

T(2n,n) gives A292692.

Cf. A258219, A258223.

Programs

Maple

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
                 + b(x-1, y+1, true, k)  ))
    end:
A:= (n, k)-> b(2*n, 0, false, k):
T:= (n, k)-> add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y>x || y<0, 0, If[x==0, 1, b[x-1, y - 1, False, k]*If[t, (x+k*y)/y, 1] + b[x-1, y+1, True, k]]]; A[n_, k_] := b[2*n, 0, False, k]; T [n_, k_] := Sum[A[n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}]/k!; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 20 2017, translated from Maple *)

Formula

T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258219(n,i).