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A271704 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*L(j,k), L the unsigned Lah numbers A271703, for n>=0 and 0<=k<=n.

%I A271704 #26 Jun 25 2025 08:03:55
%S A271704 1,0,1,0,3,1,0,11,8,1,0,49,57,15,1,0,261,424,174,24,1,0,1631,3425,
%T A271704 1930,410,35,1,0,11743,30336,21855,6320,825,48,1,0,95901,294553,
%U A271704 259161,95235,16835,1491,63,1,0,876809,3123632,3251500,1452976,325150,38864,2492,80,1
%N A271704 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*L(j,k), L the unsigned Lah numbers A271703, for n>=0 and 0<=k<=n.
%F A271704 From _Natalia L. Skirrow_, Jun 12 2025: (Start)
%F A271704 Definition can also be written Sum_{j=0..n} C(n-1, j-1)*L(j, k), where C(n, -1) = (1 if n = -1 else 0) over integer n.
%F A271704 T(n, k) = C(n-1, k-1)*A143409(n-k, k) = k*(k+1)*A271705(n+1, k+1)/(n*(n+1)) for n > 0.
%F A271704 E.g.f. for sequence b(n, k) = T(n+1, k+1): F/(1-x)^2, where F = exp(x + y*x/(1-x)) is e.g.f. of A271705.
%F A271704 E.g.f. for kth column of b: exp(x)*x^(k-1)/(1-x)^(k+1)/(k-1)!. (These cannot be integrated with x to give e.g.f.s for T(n, k) using standard functions.)
%F A271704 T(n, k) = Sum_{i=0..n-k} (n-1)_i*(i+1)*A271705(n-1-i, k-1), where (n)_i = n!/(n-i)! is the falling factorial. (End)
%e A271704 Triangle starts:
%e A271704   [1]
%e A271704   [0, 1]
%e A271704   [0, 3,     1]
%e A271704   [0, 11,    8,     1]
%e A271704   [0, 49,    57,    15,    1]
%e A271704   [0, 261,   424,   174,   24,   1]
%e A271704   [0, 1631,  3425,  1930,  410,  35,  1]
%e A271704   [0, 11743, 30336, 21855, 6320, 825, 48, 1]
%p A271704 L := (n,k) -> `if`(k<0 or k>n,0,(n-k)!*binomial(n,n-k)*binomial(n-1,n-k)):
%p A271704 T := (n,k) -> add(L(j,k)*binomial(-j,-n)*(-1)^(n-j), j=0..n):
%p A271704 seq(seq(T(n,k), k=0..n), n=0..9);
%Y A271704 A001339 (col. 1), A005563 (diag. n,n-1).
%Y A271704 Cf. A271703, A271705.
%K A271704 nonn,tabl
%O A271704 0,5
%A A271704 _Peter Luschny_, Apr 14 2016