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A258307 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258306(n,i); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

%I A258307 #16 Jun 06 2018 03:08:22
%S A258307 1,1,2,1,5,2,14,9,1,43,28,3,141,114,21,1,490,421,82,4,1785,1750,442,
%T A258307 38,1,6789,7114,1941,180,5,26809,30854,9868,1210,60,1,109632,134239,
%U A258307 46337,6191,335,6,462755,609276,235035,37321,2700,87,1,2012441,2800134,1157603,199424,15806,560,7
%N A258307 T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) *C(k,i) * A258306(n,i); triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
%H A258307 Alois P. Heinz, <a href="/A258307/b258307.txt">Rows n = 0..200, flattened</a>
%e A258307 Triangle T(n,k) begins:
%e A258307 :     1;
%e A258307 :     1;
%e A258307 :     2,     1;
%e A258307 :     5,     2;
%e A258307 :    14,     9,    1;
%e A258307 :    43,    28,    3;
%e A258307 :   141,   114,   21,    1;
%e A258307 :   490,   421,   82,    4;
%e A258307 :  1785,  1750,  442,   38,  1;
%e A258307 :  6789,  7114, 1941,  180,  5;
%e A258307 : 26809, 30854, 9868, 1210, 60, 1;
%p A258307 b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
%p A258307       `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
%p A258307                   +b(x-1, y, false, k) +b(x-1, y+1, true, k)))
%p A258307     end:
%p A258307 A:= (n, k)-> b(n, 0, false, k):
%p A258307 T:= proc(n, k) option remember;
%p A258307        add(A(n, i)*(-1)^(k-i)*binomial(k, i), i=0..k)/k!
%p A258307     end:
%p A258307 seq(seq(T(n, k), k=0..n/2), n=0..13);
%t A258307 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, (x + k*y)/y, 1] + b[x - 1, y, False, k] + b[x - 1, y + 1, True, k]]];
%t A258307 A[n_, k_] :=  b[n, 0, False, k];
%t A258307 T[n_, k_] := T[n, k] = Sum[A[n, i]*(-1)^(k-i)*Binomial[k, i], {i, 0, k}]/ k!;
%t A258307 Table[T[n, k], {n, 0, 13}, {k, 0, n/2}] // Flatten (* _Jean-François Alcover_, Jun 06 2018, from Maple *)
%Y A258307 Column k=0 gives A258312.
%Y A258307 Row sums give A258308.
%Y A258307 Cf. A258306, A258310.
%K A258307 nonn,tabf
%O A258307 0,3
%A A258307 _Alois P. Heinz_, May 25 2015