A021009 Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).
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
Offset: 0
Examples
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.
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.
- 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.
- Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
- Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See. p. 19.
- 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.
- J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
- 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
Crossrefs
Programs
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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
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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
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Mathematica
Flatten[ Table[ CoefficientList[ n!*LaguerreL[n, x], x], {n, 0, 9}]] (* Jean-François Alcover, Dec 13 2011 *) -
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
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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 */ -
PARI
row(n) = Vecrev(n!*pollaguerre(n)); \\ Michel Marcus, Feb 06 2021
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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
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
Extensions
Name changed and table given by Wolfdieter Lang, Nov 28 2011
Comments