This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A021009 #206 Feb 16 2025 08:32:34
%S A021009 1,1,-1,2,-4,1,6,-18,9,-1,24,-96,72,-16,1,120,-600,600,-200,25,-1,720,
%T A021009 -4320,5400,-2400,450,-36,1,5040,-35280,52920,-29400,7350,-882,49,-1,
%U A021009 40320,-322560,564480,-376320,117600,-18816,1568,-64,1,362880,-3265920
%N A021009 Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).
%C A021009 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
%C A021009 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
%C A021009 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
%C A021009 The unsigned triangle appears on page 83 of Ser (1933). - _N. J. A. Sloane_, Jan 16 2020
%D A021009 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.
%D A021009 G. Rota, Finite Operator Calculus, Academic Press, New York, 1975.
%D A021009 J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 83.
%H A021009 T. D. Noe, <a href="/A021009/b021009.txt">Rows n = 0..50 of triangle, flattened</a>
%H A021009 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A021009 W. A. Al-Salam, <a href="http://dx.doi.org/10.1215/S0012-7094-64-03113-8">Operational representations for the Laguerre and other polynomials</a>, Duke Math. Jour., vol 31 (1964), pp. 127-142.
%H A021009 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 5.
%H A021009 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 7.
%H A021009 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 21.
%H A021009 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 A021009 Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.
%H A021009 A. Belov-Kanel and M. Kontsevich, <a href="https://arxiv.org/abs/math/0512169">Automorphisms of the Weyl algebra</a>, arXiv preprint arXiv:0512169 [math.QA], 2005.
%H A021009 A. Belov-Kanel and M. Kontsevich, <a href="https://arxiv.org/abs/math/0512171">The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture</a>, arXiv preprint arXiv:0512171 [math.RA], 2005.
%H A021009 I. Gessel, <a href="http://www.mathe2.uni-bayreuth.de/axel/papers/gessel:applications_of_the_classical_umbral_calculus.pdf">Applications of the classical umbral calculus</a>
%H A021009 G. Hetyei, <a href="http://arxiv.org/abs/0909.4352">Meixner polynomials of the second kind and quantum algebras representing su(1,1)</a>, arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4.
%H A021009 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Janjic/janjic22.html">Some classes of numbers and derivatives</a>, JIS 12 (2009) 09.8.3.
%H A021009 Robert S. Maier, <a href="https://arxiv.org/abs/2308.10332">Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers</a>, arXiv:2308.10332 [math.CO], 2023. See. p. 19.
%H A021009 Massimo Nocentini, <a href="https://github.com/massimo-nocentini/PhD-thesis/releases/download/final-version/PhD-thesis.pdf">An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation</a>, PhD Thesis, University of Florence, 2019. See p. 31.
%H A021009 J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H A021009 J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%H A021009 M. Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Spivey/spivey31.html">On Solutions to a General Combinatorial Recurrence</a>, J. Int. Seq. 14 (2011) # 11.9.7.
%H A021009 W. Wang and T. Wang, <a href="http://dx.doi.org/10.1016/j.disc.2007.12.037">Generalized Riordan arrays</a>, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
%H A021009 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%H A021009 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A021009 a(n, m) = ((-1)^m)*n!*binomial(n, m)/m! = ((-1)^m)*((n!/m!)^2)/(n-m)! if n >= m, otherwise 0.
%F A021009 E.g.f. for m-th column: (-x/(1-x))^m /((1-x)*m!), m >= 0.
%F A021009 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
%F A021009 Sum_{m>=0} (-1)^m*a(n, m) = A002720(n). - _Philippe Deléham_, Mar 10 2004
%F A021009 E.g.f.: (1/(1-x))*exp(x*y/(x-1)). - _Vladeta Jovovic_, Apr 07 2005
%F A021009 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
%F A021009 Matrix square yields the identity matrix: L^2 = I. - _Paul D. Hanna_, Nov 22 2008
%F A021009 From _Tom Copeland_, Oct 20 2012: (Start)
%F A021009 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.
%F A021009 Then some useful relations are
%F A021009 1) (:Dx:)^n = LN(n,-:xD:) [Rodriguez formula]
%F A021009 2) (xDx)^n = x^n D^n x^n = x^n LN(n,-:xD:) [See Al-Salam ref./A132440]
%F A021009 3) (DxD)^n = D^n x^n D^n = LN(n,-:xD:) D^n [See ref. in A132440]
%F A021009 4) umbral composition LN(n,LN(.,x))= x^n [See Rota ref.]
%F A021009 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:).
%F A021009 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:).
%F A021009 An example of the umbral composition in 5) is given in A038207.
%F A021009 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)).
%F A021009 For a general discussion of umbral calculus see the Gessel link. (End)
%F A021009 From _Wolfdieter Lang_, Jan 31 2013: (Start)
%F A021009 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.
%F A021009 a(n,m) = (2*n-1)*a(n-1,m) - a(n-1,m-1) - (n-1)^2*a(n-2,m),
%F A021009 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)).
%F A021009 Simplified recurrence (using column recurrence from explicit form for a(n,m) given above):
%F A021009 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)
%F A021009 |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
%F A021009 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
%F A021009 The unsigned case is the exponential Riordan square (see A321620) of the factorial numbers. - _Peter Luschny_, Dec 06 2018
%F A021009 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
%e A021009 The triangle a(n,m) starts:
%e A021009 n\m 0 1 2 3 4 5 6 7 8
%e A021009 0: 1
%e A021009 1: 1 -1
%e A021009 2: 2 -4 1
%e A021009 3: 6 -18 9 -1
%e A021009 4: 24 -96 72 -16 1
%e A021009 5: 120 -600 600 -200 25 -1
%e A021009 6: 720 -4320 5400 -2400 450 -36 1
%e A021009 7: 5040 -35280 52920 -29400 7350 -882 49 -1
%e A021009 8:40320 -322560 564480 -376320 117600 -18816 1568 -64 1
%e A021009 ...
%e A021009 From _Wolfdieter Lang_, Jan 31 2013 (Start)
%e A021009 Recurrence (usual one): a(4,1) = 7*(-18) - 6 - 3^2*(-4) = -96.
%e A021009 Recurrence (simplified version): a(4,1) = 5*(-18) - 6 = -96.
%e A021009 Recurrence (Sage program): |a(4,1)| = 6 + 3*18 + 4*9 = 96. (End)
%e A021009 Embedded recurrence (Maple program): a(4,1) = -4!*(1 + 3) = -96.
%p A021009 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
%p A021009 # Alternative for the unsigned case (function RiordanSquare defined in A321620):
%p A021009 RiordanSquare(add(x^m, m=0..10), 10, true); # _Peter Luschny_, Dec 06 2018
%t A021009 Flatten[ Table[ CoefficientList[ n!*LaguerreL[n, x], x], {n, 0, 9}]] (* _Jean-François Alcover_, Dec 13 2011 *)
%o A021009 (Sage)
%o A021009 def A021009_triangle(dim): # computes unsigned T(n,k).
%o A021009 M = matrix(ZZ,dim,dim)
%o A021009 for n in (0..dim-1): M[n,n] = 1
%o A021009 for n in (1..dim-1):
%o A021009 for k in (0..n-1):
%o A021009 M[n,k] = M[n-1,k-1]+(2*k+1)*M[n-1,k]+(k+1)^2*M[n-1,k+1]
%o A021009 return M
%o A021009 A021009_triangle(9) # _Peter Luschny_, Sep 19 2012
%o A021009 (PARI)
%o A021009 p(n) = denominator(bestapprPade(Ser(vector(2*n, k, (k-1)!))));
%o A021009 concat(1, concat(vector(9, n, Vec(-p(n))))) \\ _Gheorghe Coserea_, Dec 01 2016
%o A021009 (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 */
%o A021009 (PARI) row(n) = Vecrev(n!*pollaguerre(n)); \\ _Michel Marcus_, Feb 06 2021
%o A021009 (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
%Y A021009 Row sums give A009940, alternating row sums are A002720.
%Y A021009 Column sequences (unsigned): A000142, A001563, A001809-A001812 for m=0..5.
%Y A021009 Central terms: A295383.
%Y A021009 For generators and generalizations see A132440.
%Y A021009 Cf. A021010, A025166, A025167, A062137, A062138, A062139, A062140, A066667, A321620.
%K A021009 sign,tabl,easy,nice
%O A021009 0,4
%A A021009 _N. J. A. Sloane_
%E A021009 Name changed and table given by _Wolfdieter Lang_, Nov 28 2011