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

%I A214299 #17 Jun 22 2015 11:38:02
%S A214299 1,2,4,24,16,80,720,288,480,3552,40320,11520,11520,28416,271104,
%T A214299 3628800,806400,576000,852480,2711040,31342080,479001600,87091200,
%U A214299 48384000,51148800,97597440,376104960,5087692800,87178291200
%N A214299 Triangle d_k(n) read by rows: number of n-th order Feynman diagrams with k interactions, 0<=k<=n.
%H A214299 Alois P. Heinz, <a href="/A214299/b214299.txt">Rows n = 0..140, flattened</a>
%H A214299 F. Battaglia, T. F. George, <a href="http://dx.doi.org/10.1007/BF01167204">A Pascal type triangle for the number of topologically distinct many-electron Feynman diagrams</a>, J. Math. Chem. 2 (1988) 241-247, triangle d_k(n).
%F A214299 d_k(n) = A214298(k)*binomial(n,k)*(2n-2k)!.
%e A214299 Triangle starts in row n=0 as:
%e A214299 1;
%e A214299 2,     4;
%e A214299 24,    16,    80;
%e A214299 720,   288,   480,   3552;
%e A214299 40320, 11520, 11520, 28416, 271104;
%p A214299 b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
%p A214299       `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, (x+y)/y, 1) +
%p A214299                    b(x-1, y+1, true)  ))
%p A214299     end:
%p A214299 T:= (n, k) -> k!*2^k*b(2*k, 0, false)*binomial(n,k)*(2*n-2*k)!:
%p A214299 seq(seq(T(n,k), k=0..n), n=0..10);  # _Alois P. Heinz_, May 23 2015
%t A214299 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_ *)
%Y A214299 Cf. A009445 (row sums), A214298 (main diagonal).
%K A214299 nonn,tabl,easy
%O A214299 0,2
%A A214299 _R. J. Mathar_, Jul 11 2012