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

0,3

Links

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

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.

Cf. A258307, A258309.

Sequence in context: A355166 A181882 A109195 * A217927 A284709 A307365

Adjacent sequences: A258307 A258308 A258309 * A258311 A258312 A258313

Keyword

nonn,tabf

Author

Alois P. Heinz, May 25 2015

Status

approved