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.

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

Author

Alois P. Heinz, May 25 2015

Keywords

• nonn
• tabf

Examples

			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;
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Column k=0 gives A001006.

T(2n,n) gives A001147.

Row sums give A258311.

Cf. A258307, A258309.

Programs

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 *)