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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.

%I A271706 #8 Jun 25 2025 11:38:59
%S A271706 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,
%T A271706 24,1,-1,1855,6489,6335,2485,441,35,1,1,14832,59332,67984,32550,7504,
%U A271706 868,48,1,-1,133497,600732,789852,445914,126126,19068,1548,63,1
%N 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.
%F A271706 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
%e A271706 Triangle starts:
%e A271706   [ 1]
%e A271706   [-1,    1]
%e A271706   [ 1,    0,    1]
%e A271706   [-1,    3,    3,    1]
%e A271706   [ 1,    8,   18,    8,    1]
%e A271706   [-1,   45,  110,   70,   15,   1]
%e A271706   [ 1,  264,  795,  640,  195,  24,  1]
%e A271706   [-1, 1855, 6489, 6335, 2485, 441, 35, 1]
%p A271706 L := (n, k) -> `if`(k<0 or k>n, 0, (n-k)!*binomial(n, n-k)*binomial(n-1, n-k)):
%p A271706 T := (n, k) -> add(L(j, k)*binomial(-j-1,-n-1), j=0..n):
%p A271706 seq(seq(T(n, k), k=0..n), n=0..9);
%p A271706 # Or:
%p A271706 T := (n, k) -> (-1)^(n-k)*binomial(n, k)*hypergeom([k-n, k], [], 1):
%p A271706 for n from 0 to 8 do seq(simplify(T(n, k)), k=0..n) od; # _Peter Luschny_, Jun 25 2025
%Y A271706 A052845 (row sums), A000240 (col. 1), A000274 (col. 2), A067998 (diag n,n-1).
%Y A271706 Cf. A271703, A271705.
%K A271706 sign,tabl
%O A271706 0,8
%A A271706 _Peter Luschny_, Apr 20 2016