0,5

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.

Take the lower triangular matrix in A049020 and invert it, then read by rows! - N. J. A. Sloane, Feb 07 2009

Exponential Riordan array [exp(x), log(1/(1-x))]. Equal to A007318*A132393. - Paul Barry, Apr 23 2009

A signed version of the triangle appears in [Gessel]. - Peter Bala, Aug 31 2012

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

From Wolfdieter Lang, Jul 28 2017: (Start)

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).

In Ismail's book the present Charlier polynomials are denoted by C_n(-x;a=1) on p. 177, eq. (6.1.25). (End)

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

T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, London, Paris, 1978, Ch. VI, 1., pp. 170-172.

Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005, EMA, Vol. 98, p. 177.

Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 6.

Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 8.

Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 22.

Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011)  11.6.7.

Ira Gessel, Congruences for Bell and Tangent numbers, The Fibonacci Quarterly, Vol. 19, Number 2, 1981.

Aoife Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.

Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.

Wolfdieter Lang, First 10 rows and more.

Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.

E.g.f.: exp(t)/(1-t)^x = Sum_{n>=0} C(x,n)*t^n/n!.

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

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.

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

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

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

From Peter Bala, Oct 23 2019: (Start)

The n-th row polynomial is

R(n, x) = Sum_{k = 0..n} (-1)^k*binomial(n, k)*k! * binomial(-x, k).

These polynomials occur in series acceleration formulas for the constant

1/e = n! * Sum_{k >= 0} (-1)^k/(k!*R(n,k)*R(n,k+1)), n >= 0. (cf. A068985, A094816 and A137346). (End)

R(n, x) = KummerU[-n, 1 - n - x, 1]. - Peter Luschny, Oct 27 2019

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

From Natalia L. Skirrow, Jun 11 2025: (Start)

G.f.: 2F0(1,y;x/(1-x)) / (1-x).

Polynomial for the n-th row is R(n,y) = 2F0(-n,y;-1).

Falling g.f. for n-th row: Sum_{k=0..n} a(n,k)*(y)_k = [x^0] 2F0(1,-n;-1/x) * (1+log(1/(1-x)))^y = [x^n] e^x * Gamma(n+1,x) * (1+log(1/(1-x)))^y, where (y)_k = y!/(y-k)! denotes the falling factorial. (End)

From Paul Barry, Apr 23 2009: (Start)

Triangle begins

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, 21, 1;

1, 2372, 5243, 4179, 1575, 301, 28, 1;

1, 16072, 38618, 34860, 15659, 3836, 518, 36, 1;

Production matrix is

1, 1;

0, 2, 1;

0, 1, 3, 1;

0, 1, 3, 4, 1;

0, 1, 4, 6, 5, 1;

0, 1, 5, 10, 10, 6, 1;

0, 1, 6, 15, 20, 15, 7, 1;

0, 1, 7, 21, 35, 35, 21, 8, 1;

0, 1, 8, 28, 56, 70, 56, 28, 9, 1; (End)

A094816 := (n, k) -> (-1)^(n-k)*add(binomial(-j-1, -n-1)*Stirling1(j, k), j=0..n):

seq(seq(A094816(n, k), k=0..n), n=0..9); # Peter Luschny, Apr 10 2016

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 *)

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 *)

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

Table[CoefficientList[p[n], x], {n, 0, 9}] // Flatten (* Peter Luschny, Oct 27 2019 *)

(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 */

(SageMath)

def a_row(n):

s = sum(binomial(n, k)*rising_factorial(x, k) for k in (0..n))

return expand(s).list()

[a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019

Columns k=0..4 give A000012, A002104, A381021, A381022, A381023.

Diagonals: A000012, A000217.

Row sums A000522, alternating row sums A024000.

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

Sequence in context: A134380 A263859 A124469 * A097712 A238688 A174117

Adjacent sequences: A094813 A094814 A094815 * A094817 A094818 A094819

nonn,easy,tabl

Philippe Deléham, Jun 12 2004

approved