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

0,4

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Feynman diagram

Formula

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

Example

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;

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 *)

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.

Sequence in context: A362296 A333668 A193293 * A158391 A016491 A347052

Adjacent sequences: A258217 A258218 A258219 * A258221 A258222 A258223

Keyword

nonn,tabl

Author

Alois P. Heinz, May 23 2015

Status

approved