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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.
5

%I #18 Jun 06 2018 03:08:14

%S 1,1,1,2,8,3,5,69,77,15,14,692,1749,890,105,42,8120,41998,41909,12039,

%T 945,132,110278,1114808,1944225,1018865,186594,10395,429,1707965,

%U 33058519,94833341,80595226,25798856,3260067,135135,1430,29750636,1093994697,4979407614,6439957299,3201618970,687652446,63390060,2027025

%N 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.

%H Alois P. Heinz, <a href="/A258223/b258223.txt">Rows n = 0..140, flattened</a>

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

%e Triangle T(n,k) begins:

%e : 1;

%e : 1, 1;

%e : 2, 8, 3;

%e : 5, 69, 77, 15;

%e : 14, 692, 1749, 890, 105;

%e : 42, 8120, 41998, 41909, 12039, 945;

%e : 132, 110278, 1114808, 1944225, 1018865, 186594, 10395;

%p b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,

%p `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)

%p + b(x-1, y+1, true, k) ))

%p end:

%p A:= (n, k)-> b(2*n, 0, false, k):

%p T:= (n, k)-> add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!:

%p seq(seq(T(n, k), k=0..n), n=0..10);

%t 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]]];

%t A[n_, k_] := b[2*n, 0, False, k];

%t T[n_, k_] := Sum[A[n, i]*(-1)^(k - i)*Binomial[k, i], {i, 0, k}]/k!;

%t Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 06 2018, from Maple *)

%Y Column k=0 gives A000108.

%Y Main diagonal gives A001147.

%Y Row sums give A258224.

%Y T(2n,n) gives A292695.

%Y Cf. A258220, A258222.

%K nonn,tabl

%O 0,4

%A _Alois P. Heinz_, May 23 2015