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.

1, 0, 1, 0, 3, 1, 0, 11, 8, 1, 0, 49, 57, 15, 1, 0, 261, 424, 174, 24, 1, 0, 1631, 3425, 1930, 410, 35, 1, 0, 11743, 30336, 21855, 6320, 825, 48, 1, 0, 95901, 294553, 259161, 95235, 16835, 1491, 63, 1, 0, 876809, 3123632, 3251500, 1452976, 325150, 38864, 2492, 80, 1

Offset: 0

Author

Peter Luschny, Apr 14 2016

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  [1]
  [0, 1]
  [0, 3,     1]
  [0, 11,    8,     1]
  [0, 49,    57,    15,    1]
  [0, 261,   424,   174,   24,   1]
  [0, 1631,  3425,  1930,  410,  35,  1]
  [0, 11743, 30336, 21855, 6320, 825, 48, 1]
		

Crossrefs

A001339 (col. 1), A005563 (diag. n,n-1).

Cf. A271703, A271705.

Programs

Maple

L := (n,k) -> `if`(k<0 or k>n,0,(n-k)!*binomial(n,n-k)*binomial(n-1,n-k)):
T := (n,k) -> add(L(j,k)*binomial(-j,-n)*(-1)^(n-j), j=0..n):
seq(seq(T(n,k), k=0..n), n=0..9);

Formula

From Natalia L. Skirrow, Jun 12 2025: (Start)

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.

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.

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.

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.)

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)