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A094816 Triangle read by rows: T(n,k), 0<=k<=n, = coefficients of Charlier polynomials: A046716 transposed. 17
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, 1, 125673, 321690, 318926, 163191, 47775, 8274, 834, 45, 1, 1, 1112083, 2995011 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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)

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..57.

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

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

P. 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

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

W. Lang, First 10 rows and more.

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.

FORMULA

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

EXAMPLE

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)

MAPLE

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

MATHEMATICA

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 *)

PROG

(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 */

CROSSREFS

Columns: A000012, A002104.

Diagonals: A000012, A000217.

Row sums A000522, alternating row sums A024000.

Sequence in context: A134380 A263859 A124469 * A097712 A238688 A174117

Adjacent sequences:  A094813 A094814 A094815 * A094817 A094818 A094819

KEYWORD

nonn,easy,tabl

AUTHOR

Philippe Deléham, Jun 12 2004

STATUS

approved

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Last modified August 17 20:59 EDT 2017. Contains 290655 sequences.