%I #44 Aug 04 2023 15:10:45

%S 1,-1,1,1,-1,1,-1,2,0,1,1,0,5,2,1,-1,9,15,15,5,1,1,35,94,85,40,9,1,-1,

%T 230,595,609,315,91,14,1,1,1624,4458,4844,2779,924,182,20,1,-1,13209,

%U 37590,43238,26817,9975,2310,330,27,1

%N T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*S1(j,k) where S1 are the Stirling cycle numbers A132393. Triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.

%C Replacing the Stirling cycle numbers in the definition by the Stirling set numbers leads to A105794.

%C From _Wolfdieter Lang_, Jun 19 2017: (Start)

%C The triangle t(n, k) = (-1)^(n-k)*T(n, k) is the matrix product of P = A007318 (Pascal) and s1 = A048994 (signed Stirling1). This is Sheffer (exp(t), log(1+t)).

%C The present triangle T is therefore the Sheffer triangle (exp(-t), -log(1-t)). Note that P is Sheffer (exp(t), t) (of the Appell type). (End)

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial">Extensions of the binomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Poisson-CharlierPolynomial.html">Poisson-Charlier Polynomial</a>

%F From _Wolfdieter Lang_, Jun 19 2017: (Start)

%F E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n,k)*x^k: exp(-t)/(1 - t)^x.

%F E.g.f. of column k sequence: exp(-x)*(-log(1-x))^k/k!, k >= 0. (End)

%F From _Peter Bala_, Oct 26 2019: (Start)

%F Let R(n, x) = (-1)^n*Sum_{k >= 0} binomial(n,k)*k!* binomial(-x,k) the n-th row polynomial of this triangle.

%F R(n, x) = c_n(-x;-1), where c_n(x;a) denotes the n-th Poisson Charlier polynomial.

%F The series representation e = Sum_{k >= 0} 1/k! is the case n = 0 of the more general result e = n!*Sum_{k >= 0} 1/(k!*R(n,k)*R(n,k+1)), n = 0,2,3,4,....

%F (End)

%F R(n, x) = KummerU(-n, 1-n-x, -1). - _Peter Luschny_, Oct 28 2019

%e Triangle starts:

%e 1;

%e -1, 1;

%e 1, -1, 1;

%e -1, 2, 0, 1;

%e 1, 0, 5, 2, 1;

%e -1, 9, 15, 15, 5, 1;

%e 1, 35, 94, 85, 40, 9, 1.

%p A269953 := (n,k) -> add(binomial(-j-1,-n-1)*abs(Stirling1(j,k)), j=0..n):

%p seq(print(seq(A269953(n, k), k=0..n)), n=0..9);

%p # Alternative:

%p egf := exp(-t)*(1-t)^(-x): ser := series(egf, t, 12): p := n -> coeff(ser, t, n):

%p seq(n!*seq(coeff(p(n), x, k), k=0..n), n=0..9); # _Peter Luschny_, Oct 28 2019

%t Flatten[Table[Sum[Binomial[-j-1,-n-1] Abs[StirlingS1[j,k]], {j,0,n}], {n,0,9},{k,0,n}]]

%t (* Or: *)

%t p [n_] := HypergeometricU[-n, 1 - n - x, -1];

%t Table[CoefficientList[p[n], x], {n, 0, 9}] (* _Peter Luschny_, Oct 28 2019 *)

%Y Cf. A000166 (row sums), A002741 (col. 1), A080956 (diag n,n-1).

%Y Cf. A105793, A105794, A132393, A007318, A048994.

%Y Cf. A001339, A046716, A082030, A095000, A137346.

%Y KummerU(-n,1-n-x,z): this sequence (z=-1), A094816 (z=1), |A137346| (z=2), A327997 (z=3).

%K sign,tabl

%O 0,8

%A _Peter Luschny_, Apr 12 2016