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A021009 Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x). 50
1, 1, -1, 2, -4, 1, 6, -18, 9, -1, 24, -96, 72, -16, 1, 120, -600, 600, -200, 25, -1, 720, -4320, 5400, -2400, 450, -36, 1, 5040, -35280, 52920, -29400, 7350, -882, 49, -1, 40320, -322560, 564480, -376320, 117600, -18816, 1568, -64, 1, 362880, -3265920 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

In absolute values, this sequence also gives the lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals (j-1)^2 (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006

A partial permutation on a set X is a bijection between two subsets of X. |T(n,n-k)| equals the numbers of partial permutations of an n-set having domain cardinality equal to k. Let E denote the operator D*x*D, where D is the derivative operator d/dx. Then E^n = sum {k = 0..n} |T(n,k)|*x^k*D^(n+k). - Peter Bala, Oct 28 2008

The unsigned triangle is the generalized Riordan array (exp(x), x) with respect to the sequence n!^2 as defined by Wang and Wang (the generalized Riordan array (exp(x), x) with respect to the sequence n! is Pascal's triangle A007318, and with respect to the sequence n!*(n+1)! is A105278). - Peter Bala, Aug 15 2013

The unsigned triangle appears on page 83 of Ser (1933). - N. J. A. Sloane, Jan 16 2020

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

Massimo Nocentini, "An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation", PhD Thesis, University of Florence, 2019. See p. 31.

G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 83.

LINKS

T. D. Noe, Rows n=0..50 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

W. A. Al-Salam, Operational representations for the Laguerre and other polynomials, Duke Math. Jour., vol 31 (1964), pp. 127-142.

Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 5.

Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 7.

Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 21.

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.

Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.

A. Belov-Kanel and M. Kontsevich, Automorphisms of the Weyl algebra, arXiv preprint arXiv:0512169 [math.QA], 2005.

A. Belov-Kanel and M. Kontsevich, The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture, arXiv preprint arXiv:0512171 [math.RA], 2005.

I. Gessel, Applications of the classical umbral calculus

G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4.

M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.

J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)

M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.

W. Wang and T. Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Eric Weisstein's World of Mathematics, Laguerre Polynomial

Index entries for sequences related to Laguerre polynomials

FORMULA

a(n, m) = ((-1)^m)*n!*binomial(n, m)/m! = ((-1)^m)*((n!/m!)^2)/(n-m)! if n >= m, otherwise 0.

E.g.f. for m-th column: (-x/(1-x))^m /((1-x)*m!), m >= 0.

Representation (of unsigned a(n, m)) as special values of Gauss hypergeometric function 2F1, in Maple notation: n!*(-1)^m*hypergeom([ -m, n+1 ], [ 1 ], 1)/m!. - Karol A. Penson, Oct 02 2003

Sum_{m>=0} (-1)^m*a(n, m) = A002720(n). - Philippe Deléham, Mar 10 2004

E.g.f.: (1/(1-x))*exp(x*y/(x-1)). - Vladeta Jovovic, Apr 07 2005

Sum_{n>=0, m>=0} a(n, m)*(x^n/n!^2)*y^m = exp(x)*BesselJ(0, 2*sqrt(x*y)). - Vladeta Jovovic, Apr 07 2005

Matrix square yields the identity matrix: L^2 = I. - Paul D. Hanna, Nov 22 2008

From Tom Copeland, Oct 20 2012: (Start)

Symbolically, with D=d/dx and LN(n,x)=n!L_n(x), define :Dx:^j = D^j x^j, :xD:^j = x^j D^j, and LN(.,x)^j = LN(j,x) = row polynomials of A021009.

Then some useful relations are

1) (:Dx:)^n = LN(n,-:xD:)    [Rodriguez formula]

2) (xDx)^n = x^n D^n x^n = x^n LN(n,-:xD:)  [See Al-Salam ref./A132440]

3) (DxD)^n = D^n x^n D^n = LN(n,-:xD:) D^n  [See ref. in A132440]

4) umbral composition LN(n,LN(.,x))= x^n   [See Rota ref.]

5) umbral comp. LN(n,-:Dx:) = LN(n,-LN(.,-:xD:)) = 2^n LN(n,-:xD:/2)= n! * (n-th row e.g.f.(x) of A038207 with x replaced by :xD:).

An example for 2) is the operator (xDx)^2 = (xDx)(xDx) = xD(x^2 + x^3D)= 2x^2 + 4x^3 D + x^4 D^2 = x^2 (2 + 4x D + x^2 D^2) = x^2 (2 + 4 :xD: + :xD:^2) = x^2 LN(2,-:xD:) = x^2 2! L_2(-:xD:).

An example of the umbral composition in 5) is given in A038207.

The op. xDx is related to the Euler/binomial transformation for power series/o.g.f.s. through exp(t*xDx) f(x) = f[x/(1-t*x)]/(1-t*x) and to the special Moebius/linear fractional/projective transformation z exp(-t*zDz)(1/z)f(z) = f(z/(1+t*z)).

For a general discussion of umbral calculus see the Gessel link. (End)

From Wolfdieter Lang, Jan 31 2013: (Start)

Standard recurrence derived from the three term recurrence of the orthogonal polynomials system {n!*L(n,x)}: L(n,x) = (2*n  - 1 - x)*L(n-1,x) - (n-1)^2*L(n-2,x), n>=1, L(-1,x) = 0, L(0,x) = 1.

  a(n,m) = (2*n-1)*a(n-1,m) - a(n-1,m-1) - (n-1)^2*a(n-2,m),

  n >=1, with a(n,-1) = 0, a(0,0) = 1, a(n,m) = 0 if n < m. (compare this with Peter Luschny's program for the unsigned case |a(n,m)| = (-1)^m*a(n,m)).

Simplified recurrence (using column recurrence from explicit form for a(n,m) given above):

a(n,m) = (n+m)*a(n-1,m) - a(n-1,m-1), n >= 1, a(0,0) = 1, a(n,-1) = 0, a(n,m) = 0 if n < m. (End)

|T(n,k)| = [x^k] (-1)^n*U(-n,1,-x), where U(a,b,x) is Kummer's hypergeometric U function. - Peter Luschny, Apr 11 2015

T(n,k) = (-1)^k*n!*S(n,k) where S(n,k) is recursively defined by: "if k = 0 then 1 else if k > n then 0 else S(n-1,k-1)/k + S(n-1,k)". - Peter Luschny, Jun 21 2017

The unsigned case is the exponential Riordan square (see A321620) of the factorial numbers. - Peter Luschny, Dec 06 2018

Omitting the diagonal and signs, this array is generated by the commutator [D^n,x^n] = D^n x^n - x^n D^n = sum_{i=0,n-1} ((n!/i!)^2/(n-i)!) x^i D^i on p. 9 of both papers by Belov-Kanel and Kontsevich. - Tom Copeland, Jan 23 2020

EXAMPLE

The triangle a(n,m) starts:

n\m   0       1      2       3      4      5    6  7  8

0:    1

1:    1      -1

2:    2      -4      1

3:    6     -18      9      -1

4:   24     -96     72     -16      1

5:  120    -600    600    -200     25     -1

6:  720   -4320   5400   -2400    450    -36    1

7: 5040  -35280  52920  -29400   7350   -882   49  -1

8:40320 -322560 564480 -376320 117600 -18816 1568 -64 1

...

From Wolfdieter Lang, Jan 31 2013 (Start)

Recurrence (usual one): a(4,1) = 7*(-18) - 6 - 3^2*(-4) = -96.

Recurrence (simplified version): a(4,1) = 5*(-18) - 6 = -96.

Recurrence (Sage program): |a(4,1)| = 6 + 3*18 + 4*9 = 96. (End)

Embedded recurrence (Maple program): a(4,1) = -4!*(1 + 3) = -96.

MAPLE

A021009 := proc(n, k) local S; S := proc(n, k) option remember; `if`(k = 0, 1, `if`( k > n, 0, S(n-1, k-1)/k + S(n-1, k))) end: (-1)^k*n!*S(n, k) end: seq(seq(A021009(n, k), k=0..n), n=0..8); # Peter Luschny, Jun 21 2017

# Alternative for the unsigned case (function RiordanSquare defined in A321620):

RiordanSquare(add(x^m, m=0..10), 10, true); # Peter Luschny, Dec 06 2018

MATHEMATICA

Flatten[ Table[ CoefficientList[ n!*LaguerreL[n, x], x], {n, 0, 9}]] (* Jean-François Alcover, Dec 13 2011 *)

PROG

(Sage)

def A021009_triangle(dim): # computes unsigned T(n, k).

    M = matrix(ZZ, dim, dim)

    for n in (0..dim-1): M[n, n] = 1

    for n in (1..dim-1):

        for k in (0..n-1):

            M[n, k] = M[n-1, k-1]+(2*k+1)*M[n-1, k]+(k+1)^2*M[n-1, k+1]

    return M

A021009_triangle(9) # Peter Luschny, Sep 19 2012

(PARI)

p(n) = denominator(bestapprPade(Ser(vector(2*n, k, (k-1)!))));

concat(1, concat(vector(9, n, Vec(-p(n)))))  \\ Gheorghe Coserea, Dec 01 2016

(PARI) {T(n, k) = if( n<0, 0, n! * polcoeff( sum(i=0, n, binomial(n, n-i) * (-x)^i / i!), k))}; /* Michael Somos, Dec 01 2016 */

(MAGMA) /* As triangle: */ [[((-1)^k)*Factorial(n)*Binomial(n, k)/Factorial(k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jan 18 2020

CROSSREFS

Row sums give A009940, alternating row sums are A002720.

Column sequences (unsigned): A000142, A001563, A001809-A001812 for m=0..5.

Central terms: A295383.

For generators and generalizations see A132440.

Cf. A021010, A025166, A025167, A062137-A062140, A066667, A321620.

Sequence in context: A204115 A204130 A204024 * A137478 A089087 A142146

Adjacent sequences:  A021006 A021007 A021008 * A021010 A021011 A021012

KEYWORD

sign,tabl,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Name changed and table given by Wolfdieter Lang, Nov 28 2011

STATUS

approved

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Last modified May 31 02:51 EDT 2020. Contains 334747 sequences. (Running on oeis4.)