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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.
6
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
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.
Sequence in context: A362296 A333668 A193293 * A158391 A016491 A347052
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 23 2015
STATUS
approved