A271706 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(-j-1, -n-1)*L(j, k), L the unsigned Lah numbers A271703, for n >= 0 and 0 <= k <= n.

1, -1, 1, 1, 0, 1, -1, 3, 3, 1, 1, 8, 18, 8, 1, -1, 45, 110, 70, 15, 1, 1, 264, 795, 640, 195, 24, 1, -1, 1855, 6489, 6335, 2485, 441, 35, 1, 1, 14832, 59332, 67984, 32550, 7504, 868, 48, 1, -1, 133497, 600732, 789852, 445914, 126126, 19068, 1548, 63, 1

Offset: 0

Author

Peter Luschny, Apr 20 2016

Keywords

• sign
• tabl

Examples

			Triangle starts:
  [ 1]
  [-1,    1]
  [ 1,    0,    1]
  [-1,    3,    3,    1]
  [ 1,    8,   18,    8,    1]
  [-1,   45,  110,   70,   15,   1]
  [ 1,  264,  795,  640,  195,  24,  1]
  [-1, 1855, 6489, 6335, 2485, 441, 35, 1]
		

Crossrefs

A052845 (row sums), A000240 (col. 1), A000274 (col. 2), A067998 (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-1,-n-1), j=0..n):
seq(seq(T(n, k), k=0..n), n=0..9);
# Or:
T := (n, k) -> (-1)^(n-k)*binomial(n, k)*hypergeom([k-n, k], [], 1):
for n from 0 to 8 do seq(simplify(T(n, k)), k=0..n) od; # Peter Luschny, Jun 25 2025

Formula

T(n, k) = (-1)^(k-n)*binomial(n, k)*hypergeom([k-n, k], [], 1). (After a formula of Natalia L. Skirrow in A271705.) - Peter Luschny, Jun 25 2025