A214299 Triangle d_k(n) read by rows: number of n-th order Feynman diagrams with k interactions, 0<=k<=n.

1, 2, 4, 24, 16, 80, 720, 288, 480, 3552, 40320, 11520, 11520, 28416, 271104, 3628800, 806400, 576000, 852480, 2711040, 31342080, 479001600, 87091200, 48384000, 51148800, 97597440, 376104960, 5087692800, 87178291200

Offset

0,2

Links

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

F. Battaglia, T. F. George, A Pascal type triangle for the number of topologically distinct many-electron Feynman diagrams, J. Math. Chem. 2 (1988) 241-247, triangle d_k(n).

Formula

d_k(n) = A214298(k)*binomial(n,k)*(2n-2k)!.

Example

Triangle starts in row n=0 as:
1;
2,     4;
24,    16,    80;
720,   288,   480,   3552;
40320, 11520, 11520, 28416, 271104;

Maple

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (x+y)/y, 1) +
                   b(x-1, y+1, true)  ))
    end:
T:= (n, k) -> k!*2^k*b(2*k, 0, false)*binomial(n, k)*(2*n-2*k)!:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, May 23 2015

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[y>x || y<0, 0, If[x == 0, 1, b[x-1, y-1, False] * If[t, (x+y)/y, 1] + b[x-1, y+1, True]]]; T[n_, k_] := k!*2^k*b[2*k, 0, False] * Binomial[n, k]*(2*n - 2*k)!; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jun 22 2015, after Alois P. Heinz *)

Crossrefs

Cf. A009445 (row sums), A214298 (main diagonal).

Sequence in context: A081680 A269573 A147761 * A090591 A098656 A138611

Adjacent sequences: A214296 A214297 A214298 * A214300 A214301 A214302

Keyword

nonn,tabl,easy

Author

R. J. Mathar, Jul 11 2012

Status

approved