A258223 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258222(n,i); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 2, 8, 3, 5, 69, 77, 15, 14, 692, 1749, 890, 105, 42, 8120, 41998, 41909, 12039, 945, 132, 110278, 1114808, 1944225, 1018865, 186594, 10395, 429, 1707965, 33058519, 94833341, 80595226, 25798856, 3260067, 135135, 1430, 29750636, 1093994697, 4979407614, 6439957299, 3201618970, 687652446, 63390060, 2027025

Offset

0,4

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258222(n,i).

Example

Triangle T(n,k) begins:
:   1;
:   1,      1;
:   2,      8,       3;
:   5,     69,      77,      15;
:  14,    692,    1749,     890,     105;
:  42,   8120,   41998,   41909,   12039,    945;
: 132, 110278, 1114808, 1944225, 1018865, 186594, 10395;

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+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, (k*x + 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, Jun 06 2018, from Maple *)

Crossrefs

Column k=0 gives A000108.

Main diagonal gives A001147.

Row sums give A258224.

T(2n,n) gives A292695.

Cf. A258220, A258222.

Sequence in context: A173823 A278115 A222296 * A154928 A201394 A169844

Adjacent sequences: A258220 A258221 A258222 * A258224 A258225 A258226

Keyword

nonn,tabl

Author

Alois P. Heinz, May 23 2015

Status

approved