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
Examples
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;
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Programs
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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 *)
Formula
T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258222(n,i).