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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.

%I A258310 #12 May 01 2022 13:37:15
%S A258310 1,1,2,1,4,3,9,14,3,21,50,15,51,204,122,15,127,784,644,105,323,3212,
%T A258310 4115,1310,105,835,13068,22587,9270,945,2188,55475,137503,85109,16764,
%U A258310 945,5798,238073,787127,614779,149754,10395
%N 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.
%H A258310 Alois P. Heinz, <a href="/A258310/b258310.txt">Rows n = 0..200, flattened</a>
%e A258310 Triangle T(n,k) begins:
%e A258310 :    1;
%e A258310 :    1;
%e A258310 :    2,     1;
%e A258310 :    4,     3;
%e A258310 :    9,    14,      3;
%e A258310 :   21,    50,     15;
%e A258310 :   51,   204,    122,    15;
%e A258310 :  127,   784,    644,   105;
%e A258310 :  323,  3212,   4115,  1310,   105;
%e A258310 :  835, 13068,  22587,  9270,   945;
%e A258310 : 2188, 55475, 137503, 85109, 16764, 945;
%p A258310 b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
%p A258310       `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
%p A258310                   +b(x-1, y, false, k) +b(x-1, y+1, true, k)))
%p A258310     end:
%p A258310 A:= (n, k)-> b(n, 0, false, k):
%p A258310 T:= proc(n, k) option remember;
%p A258310        add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!
%p A258310     end:
%p A258310 seq(seq(T(n, k), k=0..n/2), n=0..14);
%t A258310 b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0,
%t A258310      If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1]
%t A258310                  + b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]];
%t A258310 A[n_, k_] := b[n, 0, False, k];
%t A258310 T[n_, k_] := Sum[A[n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}]/k!;
%t A258310 Table[Table[T[n, k], {k, 0, n/2}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, May 01 2022, after _Alois P. Heinz_ *)
%Y A258310 Column k=0 gives A001006.
%Y A258310 T(2n,n) gives A001147.
%Y A258310 Row sums give A258311.
%Y A258310 Cf. A258307, A258309.
%K A258310 nonn,tabf
%O A258310 0,3
%A A258310 _Alois P. Heinz_, May 25 2015