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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.

%I A258223 #18 Jun 06 2018 03:08:14
%S A258223 1,1,1,2,8,3,5,69,77,15,14,692,1749,890,105,42,8120,41998,41909,12039,
%T A258223 945,132,110278,1114808,1944225,1018865,186594,10395,429,1707965,
%U A258223 33058519,94833341,80595226,25798856,3260067,135135,1430,29750636,1093994697,4979407614,6439957299,3201618970,687652446,63390060,2027025
%N 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.
%H A258223 Alois P. Heinz, <a href="/A258223/b258223.txt">Rows n = 0..140, flattened</a>
%F A258223 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258222(n,i).
%e A258223 Triangle T(n,k) begins:
%e A258223 :   1;
%e A258223 :   1,      1;
%e A258223 :   2,      8,       3;
%e A258223 :   5,     69,      77,      15;
%e A258223 :  14,    692,    1749,     890,     105;
%e A258223 :  42,   8120,   41998,   41909,   12039,    945;
%e A258223 : 132, 110278, 1114808, 1944225, 1018865, 186594, 10395;
%p A258223 b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
%p A258223       `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
%p A258223                  + b(x-1, y+1, true, k)  ))
%p A258223     end:
%p A258223 A:= (n, k)-> b(2*n, 0, false, k):
%p A258223 T:= (n, k)-> add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!:
%p A258223 seq(seq(T(n, k), k=0..n), n=0..10);
%t A258223 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 A258223 A[n_, k_] := b[2*n, 0, False, k];
%t A258223 T[n_, k_] := Sum[A[n, i]*(-1)^(k - i)*Binomial[k, i], {i, 0, k}]/k!;
%t A258223 Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 06 2018, from Maple *)
%Y A258223 Column k=0 gives A000108.
%Y A258223 Main diagonal gives A001147.
%Y A258223 Row sums give A258224.
%Y A258223 T(2n,n) gives A292695.
%Y A258223 Cf. A258220, A258222.
%K A258223 nonn,tabl
%O A258223 0,4
%A A258223 _Alois P. Heinz_, May 23 2015