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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.

%I A258220 #19 Sep 20 2017 18:51:56
%S A258220 1,1,1,4,6,1,25,49,15,1,208,498,217,28,1,2146,6016,3360,635,45,1,
%T A258220 26368,84042,56728,13997,1475,66,1,375733,1332661,1046619,316281,
%U A258220 43974,2954,91,1,6092032,23660034,21053089,7479444,1283817,114576,5334,120,1
%N 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.
%H A258220 Alois P. Heinz, <a href="/A258220/b258220.txt">Rows n = 0..140, flattened</a>
%H A258220 Wikipedia, <a href="https://en.wikipedia.org/wiki/Feynman_diagram">Feynman diagram</a>
%F A258220 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258219(n,i).
%e A258220 Triangle T(n,k) begins:
%e A258220 :     1;
%e A258220 :     1,     1;
%e A258220 :     4,     6,     1;
%e A258220 :    25,    49,    15,     1;
%e A258220 :   208,   498,   217,    28,    1;
%e A258220 :  2146,  6016,  3360,   635,   45,  1;
%e A258220 : 26368, 84042, 56728, 13997, 1475, 66, 1;
%p A258220 b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
%p A258220       `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
%p A258220                  + b(x-1, y+1, true, k)  ))
%p A258220     end:
%p A258220 A:= (n, k)-> b(2*n, 0, false, k):
%p A258220 T:= (n, k)-> add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!:
%p A258220 seq(seq(T(n, k), k=0..n), n=0..10);
%t A258220 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 *)
%Y A258220 Column k=0 gives A005411 (for n>0).
%Y A258220 Main diagonal and lower diagonal give: A000012, A000384(n+1).
%Y A258220 Row sums give A258221.
%Y A258220 T(2n,n) gives A292692.
%Y A258220 Cf. A258219, A258223.
%K A258220 nonn,tabl
%O A258220 0,4
%A A258220 _Alois P. Heinz_, May 23 2015