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%I #128 Aug 09 2022 07:08:25
%S 1,1,1,1,3,1,1,8,6,1,1,24,29,10,1,1,89,145,75,15,1,1,415,814,545,160,
%T 21,1,1,2372,5243,4179,1575,301,28,1,1,16072,38618,34860,15659,3836,
%U 518,36,1,1,125673,321690,318926,163191,47775,8274,834,45,1,1,1112083,2995011
%N Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.
%C The a-sequence for this Sheffer matrix is A027641(n)/A027642(n) (Bernoulli numbers) and the z-sequence is A130189(n)/ A130190(n). See the W. Lang link.
%C Take the lower triangular matrix in A049020 and invert it, then read by rows! - _N. J. A. Sloane_, Feb 07 2009
%C Exponential Riordan array [exp(x), log(1/(1-x))]. Equal to A007318*A132393. - _Paul Barry_, Apr 23 2009
%C A signed version of the triangle appears in [Gessel]. - _Peter Bala_, Aug 31 2012
%C T(n,k) is the number of permutations over all subsets of {1,2,...,n} (Cf. A000522) that have exactly k cycles. T(3,2) = 6: We permute the elements of the subsets {1,2}, {1,3}, {2,3}. Each has one permutation with 2 cycles. We permute the elements of {1,2,3} and there are three permutations that have 2 cycles. 3*1 + 1*3 = 6. - _Geoffrey Critzer_, Feb 24 2013
%C From _Wolfdieter Lang_, Jul 28 2017: (Start)
%C In Chihara's book the row polynomials (with rising powers) are the Charlier polynomials (-1)^n*C^(a)_n(-x), with a = -1, n >= 0. See p. 170, eq. (1.4).
%C In Ismail's book the present Charlier polynomials are denoted by C_n(-x;a=1) on p. 177, eq. (6.1.25). (End)
%D T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, London, Paris, 1978, Ch. VI, 1., pp. 170-172.
%D Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005, EMA, Vol. 98, p. 177.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry3/barry100r.html">The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms</a>, J. Int. Seq. 13 (2010) # 10.8.4, example 6.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry4/barry122.html">Exponential Riordan Arrays and Permutation Enumeration</a>, J. Int. Seq. 13 (2010) # 10.9.1, example 8.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry1/barry97r2.html">Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms</a>, J. Int. Seq. 14 (2011) # 11.2.2, example 22.
%H Paul Barry, <a href="http://arxiv.org/abs/1105.3044">Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays</a>, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry5/barry112.html">J. Int. Seq. 14 (2011) 11.6.7</a>.
%H Ira Gessel, <a href="https://www.fq.math.ca/Scanned/19-2/gessel.pdf">Congruences for Bell and Tangent numbers</a>, The Fibonacci Quarterly, Vol. 19, Number 2, 1981.
%H Aoife Hennessy and Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry6/barry161.html">Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials</a>, J. Int. Seq. 14 (2011) # 11.8.2.
%H Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H Wolfdieter Lang, <a href="/A094816/a094816.txt">First 10 rows and more.</a>
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%H W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa35/aa3511.pdf">Arithmetic properties of Bell numbers to a composite modulus I</a>, Acta Arithmetica 35 (1979), pp. 1-16.
%F E.g.f.: exp(t)/(1-t)^(-x) = Sum_{n>=0} C(-x,n)*t^n/n!.
%F Sum_{k = 0..n} T(n, k)*x^k = C(x, n), Charlier polynomials; C(x, n)= A024000(n), A000012(n), A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n), A095722(n), A095740(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - _Philippe Deléham_, Feb 27 2013
%F T(n+1, k) = (n+1)*T(n, k) + T(n, k-1) - n*T(n-1, k) with T(0, 0) = 1, T(0, k) = 0 if k>0, T(n, k) = 0 if k<0.
%F PS*A008275*PS as infinite lower triangular matrices, where PS is a triangle with PS(n, k) = (-1)^k*A007318(n, k). PS = 1/PS. - _Gerald McGarvey_, Aug 20 2009
%F T(n,k) = (-1)^(n-k)*Sum_{j=0..n} C(-j-1, -n-1)*S1(j, k) where S1 are the signed Stirling numbers of the first kind. - _Peter Luschny_, Apr 10 2016
%F Absolute values T(n,k) of triangle (-1)^(n+k) T(n,k) where row n gives coefficients of x^k, 0 <= k <= n, in expansion of Sum_{k=0..n} binomial(n,k) (-1)^(n-k) x^{(k)}, where x^{(k)} := Product_{i=0..k-1} (x-i), k >= 1, and x^{(0)} := 1, the falling factorial powers. - _Daniel Forgues_, Oct 13 2019
%F From _Peter Bala_, Oct 23 2019: (Start)
%F The n-th row polynomial is
%F R(n, x) = Sum_{k = 0..n} (-1)^k*binomial(n, k)*k! * binomial(-x, k).
%F These polynomials occur in series acceleration formulas for the constant
%F 1/e = n! * Sum_{k >= 0} (-1)^k/(k!*R(n,k)*R(n,k+1)), n >= 0. (cf. A068985, A094816 and A137346). (End)
%F R(n, x) = KummerU[-n, 1 - n - x, 1]. - _Peter Luschny_, Oct 27 2019
%F Sum_{j=0..m} (-1)^(m-j) * Bell(n+j) * T(m,j) = m! * Sum_{k=0..n} binomial(k,m) * Stirling2(n,k). - _Vaclav Kotesovec_, Aug 06 2021
%e From _Paul Barry_, Apr 23 2009: (Start)
%e Triangle begins
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 8, 6, 1;
%e 1, 24, 29, 10, 1;
%e 1, 89, 145, 75, 15, 1;
%e 1, 415, 814, 545, 160, 21, 1;
%e 1, 2372, 5243, 4179, 1575, 301, 28, 1;
%e 1, 16072, 38618, 34860, 15659, 3836, 518, 36, 1;
%e Production matrix is
%e 1, 1;
%e 0, 2, 1;
%e 0, 1, 3, 1;
%e 0, 1, 3, 4, 1;
%e 0, 1, 4, 6, 5, 1;
%e 0, 1, 5, 10, 10, 6, 1;
%e 0, 1, 6, 15, 20, 15, 7, 1;
%e 0, 1, 7, 21, 35, 35, 21, 8, 1;
%e 0, 1, 8, 28, 56, 70, 56, 28, 9, 1; (End)
%p A094816 := (n,k) -> (-1)^(n-k)*add(binomial(-j-1,-n-1)*Stirling1(j,k), j=0..n):
%p seq(seq(A094816(n, k), k=0..n), n=0..9); # _Peter Luschny_, Apr 10 2016
%t nn=10;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[Series[ Exp[x]/(1-x)^y,{x,0,nn}],{x,y}]]//Grid (* _Geoffrey Critzer_, Feb 24 2013 *)
%t Flatten[Table[(-1)^(n-k) Sum[Binomial[-j-1,-n-1] StirlingS1[j,k],{j,0,n}], {n,0,9},{k,0,n}]] (* _Peter Luschny_, Apr 10 2016 *)
%t p[n_] := HypergeometricU[-n, 1 - n - x, 1];
%t Table[CoefficientList[p[n], x], {n,0,9}] // Flatten (* _Peter Luschny_, Oct 27 2019 *)
%o (PARI) {T(n, k)= local(A); if( k<0 || k>n, 0, A = x * O(x^n); polcoeff( n! * polcoeff( exp(x + A) / (1 - x + A)^y, n), k))} /* _Michael Somos_, Nov 19 2006 */
%o (Sage)
%o def a_row(n):
%o s = sum(binomial(n,k)*rising_factorial(x,k) for k in (0..n))
%o return expand(s).list()
%o [a_row(n) for n in (0..9)] # _Peter Luschny_, Jun 28 2019
%Y Columns: A000012, A002104.
%Y Diagonals: A000012, A000217.
%Y Row sums A000522, alternating row sums A024000.
%Y KummerU(-n,1-n-x,z): this sequence (z=1), |A137346| (z=2), A327997 (z=3).
%K nonn,easy,tabl
%O 0,5
%A _Philippe Deléham_, Jun 12 2004
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