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A258310
T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258309(n,i); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
4
1, 1, 2, 1, 4, 3, 9, 14, 3, 21, 50, 15, 51, 204, 122, 15, 127, 784, 644, 105, 323, 3212, 4115, 1310, 105, 835, 13068, 22587, 9270, 945, 2188, 55475, 137503, 85109, 16764, 945, 5798, 238073, 787127, 614779, 149754, 10395
OFFSET
0,3
LINKS
EXAMPLE
Triangle T(n,k) begins:
: 1;
: 1;
: 2, 1;
: 4, 3;
: 9, 14, 3;
: 21, 50, 15;
: 51, 204, 122, 15;
: 127, 784, 644, 105;
: 323, 3212, 4115, 1310, 105;
: 835, 13068, 22587, 9270, 945;
: 2188, 55475, 137503, 85109, 16764, 945;
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, (k*x+y)/y, 1)
+b(x-1, y, false, k) +b(x-1, y+1, true, k)))
end:
A:= (n, k)-> b(n, 0, false, k):
T:= proc(n, k) option remember;
add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!
end:
seq(seq(T(n, k), k=0..n/2), n=0..14);
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, (k*x + y)/y, 1]
+ b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]];
A[n_, k_] := b[n, 0, False, k];
T[n_, k_] := Sum[A[n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}]/k!;
Table[Table[T[n, k], {k, 0, n/2}], {n, 0, 14}] // Flatten (* Jean-François Alcover, May 01 2022, after Alois P. Heinz *)
CROSSREFS
Column k=0 gives A001006.
T(2n,n) gives A001147.
Row sums give A258311.
Sequence in context: A355166 A181882 A109195 * A217927 A284709 A307365
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 25 2015
STATUS
approved