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A269953 Triangle read by rows: T(n, k) = Sum_{j=0..n} binomial(-j-1, -n-1)*S1(j, k) where S1 are the Stirling cycle numbers A132393.

%I A269953 #61 Feb 25 2025 22:55:24
%S A269953 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 A269953 230,595,609,315,91,14,1,1,1624,4458,4844,2779,924,182,20,1,-1,13209,
%U A269953 37590,43238,26817,9975,2310,330,27,1
%N A269953 Triangle read by rows: T(n, k) = Sum_{j=0..n} binomial(-j-1, -n-1)*S1(j, k) where S1 are the Stirling cycle numbers A132393.
%C A269953 Replacing the Stirling cycle numbers in the definition by the Stirling set numbers leads to A105794.
%C A269953 From _Wolfdieter Lang_, Jun 19 2017: (Start)
%C A269953 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 A269953 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)
%C A269953 The triangle T(n,k) is a representative of the parametric family of triangles T(m,n,k), whose columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=1. See A381082. - _Igor Victorovich Statsenko_, Feb 14 2025
%H A269953 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial">Extensions of the binomial</a>
%H A269953 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Poisson-CharlierPolynomial.html">Poisson-Charlier Polynomial</a>
%F A269953 From _Wolfdieter Lang_, Jun 19 2017: (Start)
%F A269953 E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n,k)*x^k: exp(-t)/(1 - t)^x.
%F A269953 E.g.f. of column k sequence: exp(-x)*(-log(1-x))^k/k!, k >= 0. (End)
%F A269953 From _Peter Bala_, Oct 26 2019: (Start)
%F A269953 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 A269953 R(n, x) = c_n(-x;-1), where c_n(x;a) denotes the n-th Poisson Charlier polynomial.
%F A269953 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,.... (End)
%F A269953 R(n, x) = KummerU(-n, 1-n-x, -1). - _Peter Luschny_, Oct 28 2019
%e A269953 Triangle starts:
%e A269953    1;
%e A269953   -1,  1;
%e A269953    1, -1,  1;
%e A269953   -1,  2,  0,  1;
%e A269953    1,  0,  5,  2,  1;
%e A269953   -1,  9, 15, 15,  5,  1;
%e A269953    1, 35, 94, 85, 40,  9,  1.
%p A269953 A269953 := (n,k) -> add(binomial(-j-1,-n-1)*abs(Stirling1(j,k)), j=0..n):
%p A269953 seq(print(seq(A269953(n, k), k=0..n)), n=0..9);
%p A269953 # Alternative:
%p A269953 egf := exp(-t)*(1-t)^(-x): ser := series(egf, t, 12): p := n -> coeff(ser, t, n):
%p A269953 seq(n!*seq(coeff(p(n), x, k), k=0..n), n=0..9); # _Peter Luschny_, Oct 28 2019
%t A269953 Flatten[Table[Sum[Binomial[-j-1,-n-1] Abs[StirlingS1[j,k]], {j,0,n}], {n,0,9},{k,0,n}]]
%t A269953 (* Or: *)
%t A269953 p [n_] := HypergeometricU[-n, 1 - n - x, -1];
%t A269953 Table[CoefficientList[p[n], x], {n, 0, 9}] (* _Peter Luschny_, Oct 28 2019 *)
%Y A269953 Columns k=0..4 give A033999, A002741, A381064, A381065, A381066.
%Y A269953 Cf. A000166 (row sums), A080956 (diag n,n-1).
%Y A269953 Cf. A105793, A105794, A132393, A007318, A048994.
%Y A269953 Cf. A001339, A046716, A082030, A095000, A137346.
%Y A269953 KummerU(-n,1-n-x,z): this sequence (z=-1), A094816 (z=1), |A137346| (z=2), A327997 (z=3).
%K A269953 sign,tabl
%O A269953 0,8
%A A269953 _Peter Luschny_, Apr 12 2016