A062139 -id:A062139 - cp's OEIS frontend

A062193 Fourth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

1, 24, 420, 6720, 105840, 1693440, 27941760, 479001600, 8562153600, 159826867200, 3116623910400, 63465795993600, 1348648164864000, 29877743960064000, 689322235650048000, 16543733655601152000, 412559358036553728000, 10678006913887272960000, 286526518855975157760000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials.

Cf. A001710, A005990, A005461, A062139.

Cf. A001113, A001620, A091725, A099285.

[Factorial(n+3)*binomial(n+5, 5)/Factorial(3): n in [0..30]]; // G. C. Greubel, May 11 2018

With[{nn=20},CoefficientList[Series[(1+15*x+30*x^2+10*x^3)/(1-x)^9, {x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 02 2018 *)

{ f=2; for (n=0, 100, f*=n + 3; write("b062193.txt", n, " ", f*binomial(n + 5, 5)/6) ) } \\ Harry J. Smith, Aug 02 2009

my(x='x+O('x^30)); Vec(serlaplace((1+15*x+30*x^2+10*x^3)/(1-x)^9)) \\ G. C. Greubel, May 11 2018

[binomial(n,5)*factorial (n-2)/6 for n in range(5, 21)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: (1+15*x+30*x^2+10*x^3)/(1-x)^9.
a(n) = A062139(n+3, 3).
a(n) = (n+3)!*binomial(n+5, 5)/3!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n-3) = (-1)^(n-1)*f(n,3,-6), (n>=3). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 75*(Ei(1) - gamma) - 30*e - 65/4, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=0} (-1)^n/a(n) = 315*(gamma - Ei(-1)) - 180/e - 735/4, where Ei(-1) = -A099285. (End)

A062194 Fifth column sequence of triangle A062139 (generalized a=2 Laguerre).

Original entry on oeis.org

1, 35, 840, 17640, 352800, 6985440, 139708800, 2854051200, 59935075200, 1298593296000, 29088489830400, 674324082432000, 16183777978368000, 402104637462528000, 10339833534750720000, 275039572024369152000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials.

Cf. A001710, A005461, A005990, A062193.

Cf. A001113, A001620, A091725, A099285.

List([0..15],n->Factorial(n+4)*Binomial(n+6,6)/Factorial(4)); # Muniru A Asiru, Jul 01 2018

[Factorial(n+4)*Binomial(n+6, 6)/Factorial(4): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+4)!*Binomial[n+6,6]/4!, {n, 0, 20}] (* G. C. Greubel, May 12 2018 *)

{ f=6; for (n=0, 100, f*=n + 4; write("b062194.txt", n, " ", f*binomial(n + 6, 6)/24) ) } \\ Harry J. Smith, Aug 02 2009

[binomial(n,6)*factorial (n-2)/factorial (4) for n in range(6, 22)] # Zerinvary Lajos, Jul 07 2009

E.g.f.: (1 + 24*x + 90*x^2 + 80*x^3 + 15*x^4)/(1-x)^11.
a(n) = A062139(n+4, 4).
a(n) = (n+4)!*binomial(n+6, 6)/4!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..n} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-4) = (-1)^n*f(n,4,-7), (n >= 4). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 336*(gamma - Ei(1)) - 96*e + 3524/5, where gamma = A001620, Ei(1) = A091725, and e = A001113.
Sum_{n>=0} (-1)^n/a(n) = 3264*(gamma - Ei(-1)) - 1920/e - 9464/5, where Ei(-1) = -A099285. (End)

A062195 Sixth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

Original entry on oeis.org

1, 48, 1512, 40320, 997920, 23950080, 570810240, 13699445760, 333923990400, 8310997094400, 211930425907200, 5548723878297600, 149353151057510400, 4135933413900288000, 117874102296158208000

Offset: 0

Wolfdieter Lang, Jun 19 2001

Harry J. Smith, Table of n, a(n) for n = 0..100
Index entries for sequences related to Laguerre polynomials.

Cf. A001710, A005990, A005461, A062139, A062193, A062194.

Cf. A001113, A001620, A091725, A099285.

[Factorial(n+5)*Binomial(n+7, 7)/Factorial(5): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+5)!*Binomial[n+7, 7]/5!, {n, 0, 20}] (* G. C. Greubel, May 12 2018 *)

{ f=24; for (n=0, 100, f*=n + 5; write("b062195.txt", n, " ", f*binomial(n + 7, 7)/120) ) } \\ Harry J. Smith, Aug 02 2009

E.g.f.: N(2;5, x)/(1-x)^13 with N(2;5, x) := Sum_{k=0..5} A062196(5, k)*x^k = 1+35*x+210*x^2+350*x^3+175*x^4+21*x^5.
a(n) = A062139(n+5, 5).
a(n) = (n+5)!*binomial(n+7, 7)/5!.
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j) * Stirling1(n,k) * Stirling2(j,i) * x^(k-j) then a(n-5) = (-1)^(n-1)*f(n,5,-8), (n>=5). - Milan Janjic, Mar 01 2009
From Amiram Eldar, May 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 1295*(Ei(1) - gamma) + 2170*e - 22813/3, where Ei(1) = A091725, gamma = A001620, and e = A001113.
Sum_{n>=0} (-1)^n/a(n) = 36575*(gamma - Ei(-1)) - 21700/e - 63455/3, where Ei(-1) = -A099285. (End)

A062197 Row sums of signed triangle A062139 (generalized a=2 Laguerre).

Original entry on oeis.org

1, 2, 5, 14, 37, 34, -887, -14050, -168919, -1916542, -21607859, -245387858, -2799384755, -31558843486, -337767590383, -3063846770626, -11912361112367, 477367592119810, 21032925955607701, 627398853149961038, 16703816669710968821

Offset: 0

Wolfdieter Lang, Jun 19 2001

G. C. Greubel, Table of n, a(n) for n = 0..448
Index entries for sequences related to Laguerre polynomials

Cf. A062139.

[Factorial(n)*(&+[(-1)^k*Binomial(n+2, n-k)/Factorial(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, May 13 2018

a := n -> (n+2)!*hypergeom([-n],[3],1)/2:
seq(simplify(a(n)), n=0..20); # Peter Luschny, Apr 11 2015

Table[n!*LaguerreL[n, 2, 1],{n,0,20}] (* Vaclav Kotesovec, Aug 01 2013 *)

for(n=0,30, print1(n!*sum(k=0,n, (-1)^k*binomial(n+2, n-k)/k!), ", ")) \\ G. C. Greubel, May 13 2018

a(n) = vecsum(Vec(n!*pollaguerre(n, 2))); \\ Michel Marcus, Feb 06 2021

E.g.f.: exp(-x/(1-x))/(1-x)^3.
a(n) = Sum_{m=0..n} ((-1)^m)*n!*binomial(n+2, n-m)/m!.
a(n) = 2*n*a(n-1) - (n-1)*(n+1)*a(n-2). - Vaclav Kotesovec, Aug 01 2013
a(n) = (n+2)!*hypergeom([-n],[3],1)/2. - Peter Luschny, Apr 11 2015

A021009 Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).

Original entry on oeis.org

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

N. J. A. Sloane

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

			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.

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.

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

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, A062138, A062139, A062140, A066667, A321620.

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

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

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

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

{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 */

row(n) = Vecrev(n!*pollaguerre(n)); \\ Michel Marcus, Feb 06 2021

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

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

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

A052852 Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)).

Original entry on oeis.org

0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561, 175721140, 2581284541, 40864292184, 693347907421, 12548540320876, 241253367679185, 4909234733857696, 105394372192969489, 2380337795595885156, 56410454014314490981, 1399496554158060983080

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple grammar.
Number of {121,212}-avoiding n-ary words of length n. - Ralf Stephan, Apr 20 2004
The infinite continued fraction (1+n)/(1+(2+n)/(2+(3+n)/(3+...))) converges to the rational number A052852(n)/A000262(n) when n is a positive integer. - David Angell (angell(AT)maths.unsw.edu.au), Dec 18 2008
a(n) is the total number of components summed over all nilpotent partial permutations of [n]. - Geoffrey Critzer, Feb 19 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David Angell, A family of continued fractions, Journal of Number Theory, Volume 130, Issue 4, April 2010, Pages 904-911, Section 2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 820
Florent Hivert, Jean-Christophe Novelli and Jean-Yves Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.
Index entries for sequences related to Laguerre polynomials

Row sums of unsigned triangle A062139 (generalized a=2 Laguerre).
Cf. A000262. A000169, A000312.
Cf. A059114, A288268.

[n eq 0 select 0 else Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 0), -1): n in [0..30]]; // G. C. Greubel, Feb 23 2021

spec := [S,{B=Set(C),C=Sequence(Z,1 <= card),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a := n -> ifelse(n = 0, 0, n!*hypergeom([-n+1], [1], -1)): seq(simplify(a(n)), n = 0..18);  # Peter Luschny, Dec 30 2024

Table[n!*SeriesCoefficient[(x/(1-x))*E^(x/(1-x)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 09 2012 *)
Table[If[n==0, 0, n!*LaguerreL[n-1, 0, -1]], {n, 0, 30}] (* G. C. Greubel, Feb 23 2021 *)

my(x='x+O('x^30)); concat([0], Vec(serlaplace((x/(1-x))*exp(x/(1-x))))) \\ G. C. Greubel, May 15 2018

[0 if n==0 else factorial(n)*gen_laguerre(n-1, 0, -1) for n in (0..30)] # G. C. Greubel, Feb 23 2021

D-finite with recurrence: a(1)=1, a(0)=0, (n^2+2*n)*a(n)+(-4-2*n)*a(n+1)+ a(n+2)=0.
a(n) = Sum_{m=0..n} n!*binomial(n+2, n-m)/m!. - Wolfdieter Lang, Jun 19 2001
a(n) = n*A002720(n-1). [Riordan] - Vladeta Jovovic, Mar 18 2005
Related to an n-dimensional series: for n>=1, a(n) = (n!/e)*Sum_{k_n>=k_{n-1}>=...>=k_1>=0} 1/(k_n)!. - Benoit Cloitre, Sep 30 2006
E.g.f.: (x/(1-x))*exp((x/(1-x))) = (x/(1-x))*G(0); G(k)=1+x/((2*k+1)*(1-x)-x*(1-x)*(2*k+1)/(x+(1-x)*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
a(n) = D^n(x*exp(x)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A000262 and A005493. - Peter Bala, Nov 25 2011
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+1/4)/sqrt(2). - Vaclav Kotesovec, Oct 09 2012
a(n) = n!*hypergeom([-n+1], [1], -1) for n>=1. - Peter Luschny, Oct 18 2014 [Simplified by Natalia L. Skirrow, 30 December 2024]
a(n) = Sum_{k=0..n} L(n,k)*k; L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014
a(n) = n!*LaguerreL(n-1, 0, -1) for n>=1. - Peter Luschny, Apr 08 2015, simplified Dec 30 2024
The series reversion of the e.g.f. equals W(x)/(1 + W(x)) = x - 2^2*x^2/2! + 3^3*x^3/3! - 4^4*x^4/4! + ..., essentially the e.g.f. for a signed version of A000312, where W(x) is Lambert's W-function (see A000169). - Peter Bala, Jun 14 2016
Equals column A059114(n, 2) for n >= 1. - G. C. Greubel, Feb 23 2021
a(n) = Sum_{k=1..n} k * A271703(n,k). - Geoffrey Critzer, Feb 19 2022

A062196 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).

Original entry on oeis.org

1, 1, 3, 1, 8, 6, 1, 15, 30, 10, 1, 24, 90, 80, 15, 1, 35, 210, 350, 175, 21, 1, 48, 420, 1120, 1050, 336, 28, 1, 63, 756, 2940, 4410, 2646, 588, 36, 1, 80, 1260, 6720, 14700, 14112, 5880, 960, 45, 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55

Offset: 0

Wolfdieter Lang, Jun 19 2001

Also the coefficient triangle of certain polynomials N(2; m,x) := Sum_{k=0..m} T(m,k)*x^k. The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=2) Laguerre triangle L(2; n+m,m) = A062139(n+m,m), n >= 0, is N(2; m,x)/(1-x)^(3+2*m), with the row polynomials N(2; m,x).

			Triangle starts:
  n\k 0...1.....2......3..... 4.....;
  [0] 1;
  [1] 1,  3;
  [2] 1,  8,    6;
  [3] 1, 15,   30,    10;
  [4] 1, 24,   90,    80,    15;
  [5] 1, 35,  210,   350,   175,    21;
  [6] 1, 48,  420,  1120,  1050,   336,    28;
  [7] 1, 63,  756,  2940,  4410,  2646,   588,    36;
  [8] 1, 80, 1260,  6720, 14700, 14112,  5880,   960,   45;
  [9] 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), this sequence (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Sums include: A001791 (row), (-1)^n*A089849(n+1) (alternating sign row).
Diagonals: A000217 (k=n), A002417 (k=n-1), A001297 (k=n-2), A105946 (k=n-3), A105947 (k=n-4), A105948 (k=n-5), A107319 (k=n-6).
Columns: A005563 (k=1), A033487 (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).
Cf. A000108, A089849.

A062196:= func;
[A062196(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 21 2025

T := (n, k) -> binomial(n, k)*binomial(n + 2, k);
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Sep 30 2021

A062196[n_, k_]:= Binomial[n, k]*Binomial[n+2, k];
Table[A062196[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 21 2025 *)

def A062196(n,k): return binomial(n,k)*binomial(n+2,k)
print(flatten([[A062196(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Feb 21 2025

T(m, k) = [x^k] N(2; m, x), where N(2; m, x) = ((1-x)^(3+2*m))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+3))).
N(2; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+2-j)!/((m+2)!*(m-j)!)*(x^(m-j)))*(1-x)^j).
T(n,m) = binomial(n, m)*(binomial(n+1, m) + binomial(n+1, m-1)). - Vladimir Kruchinin, Apr 06 2018
From G. C. Greubel, Feb 21 2025: (Start)
T(2*n, n) = (n+1)^2*A000108(n)*A000108(n+1).
T(2*n-1, n) = (4*n^2 - 1)*A000108(n-1)*A000108(n), n >= 1.
T(2*n+1, n) = (1/2)*binomial(n+2,2)*A000108(n+1)*A000108(n+2). (End)

New name by Peter Luschny, Sep 30 2021

A066667 Coefficient triangle of generalized Laguerre polynomials (a=1).

Original entry on oeis.org

1, 2, -1, 6, -6, 1, 24, -36, 12, -1, 120, -240, 120, -20, 1, 720, -1800, 1200, -300, 30, -1, 5040, -15120, 12600, -4200, 630, -42, 1, 40320, -141120, 141120, -58800, 11760, -1176, 56, -1, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016

Offset: 0

Christian G. Bower, Dec 17 2001

Same as A008297 (Lah triangle) except for signs.
Row sums give A066668. Unsigned row sums give A000262.
The Laguerre polynomials L(n;x;a=1) under discussion are connected with Hermite-Bell polynomials p(n;x) for power -1 (see also A215216) defined by the following relation: p(n;x) := x^(2n)*exp(x^(-1))*(d^n exp(-x^(-1))/dx^n). We have L(n;x;a=1)=(-x)^(n-1)*p(n;1/x), p(n+1;x)=x^2(dp(n;x)/dx)+(1-2*n*x)p(n;x), and p(1;x)=1, p(2;x)=1-2*x, p(3;x)=1-6*x+6*x^2, p(4;x)=1-12*x+36*x^2-24*x^3, p(5;x)=1-20*x+120*x^2-240*x^3+120*x^4. Note that if we set w(n;x):=x^(2n)*p(n;1/x) then w(n+1;x)=(w(n;x)-(dw(n;x)/dx))*x^2, which is almost analogous to the recurrence formula for Bell polynomials B(n+1;x)=(B(n;x)+(dB(n;x)/dx))*x. - Roman Witula, Aug 06 2012.

			Triangle a(n,m) begins
n\m     0        1       2       3      4      5    6   7  8
0:      1
1:      2       -1
2:      6       -6       1
3:     24      -36      12      -1
4:    120     -240     120     -20      1
5:    720    -1800    1200    -300     30     -1
6:   5040   -15120   12600   -4200    630    -42    1
7:  40320  -141120  141120  -58800  11760  -1176   56  -1
8: 362880 -1451520 1693440 -846720 211680 -28224 2016 -72  1
9: 3628800, -16329600, 21772800, -12700800, 3810240, -635040, 60480, -3240, 90, -1.
Reformatted and extended by _Wolfdieter Lang_, Jan 31 2013.
From _Wolfdieter Lang_, Jan 31 2013 (Start)
Recurrence (standard): a(4,2) = 2*4*12 - (-36) - 4*3*1 = 120.
Recurrence (simple): a(4,2) = 7*12 - (-36) = 120. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 778 (22.5.17).
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 95 (4.1.62)
R. Witula, E. Hetmaniok, and D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 >= n >= 150, 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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 778 (22.5.17).
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A021009, A062137, A062138, A062139, A062140, A089231.

A066667 := (n, k) -> (-1)^k*binomial(n, k)*(n + 1)!/(k + 1)!:
for n from 0 to 9 do seq(A066667(n,k), k = 0..n) od; # Peter Luschny, Jun 22 2022

Table[(-1)^m*Binomial[n, m]*(n + 1)!/(m + 1)!, {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Sep 04 2019 *)

row(n) = Vecrev(n!*pollaguerre(n, 1)); \\ Michel Marcus, Feb 06 2021

E.g.f. (relative to x, keep y fixed): A(x)=(1/(1-x))^2*exp(x*y/(x-1)).
From Wolfdieter Lang, Jan 31 2013: (Start)
a(n,m) = (-1)^m*binomial(n,m)*(n+1)!/(m+1)!, n >= m >= 0. [corrected by Georg Fischer, Oct 26 2022]
Recurrence from standard three term recurrence for orthogonal generalized Laguerre polynomials {P(n,x):=n!*L(1,n,x)}:
  P(n,x) = (2*n-x)*P(n-1,x) - n*(n-1)*P(n-2), n>=1, P(-1,x) = 0, P(0,x) = 1.
  a(n,m) = 2*n*a(n-1,m) - a(n-1,m-1) - n*(n-1)*a(n-2,m), n>=1, a(0,0) =1, a(n,-1) = 0, a(n,m) = 0 if n < m.
Simplified recurrence from explicit form of a(n,m):
  a(n,m) = (n+m+1)*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)

A062137 Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x).

Original entry on oeis.org

1, 4, -1, 20, -10, 1, 120, -90, 18, -1, 840, -840, 252, -28, 1, 6720, -8400, 3360, -560, 40, -1, 60480, -90720, 45360, -10080, 1080, -54, 1, 604800, -1058400, 635040, -176400, 25200, -1890, 70, -1, 6652800, -13305600, 9313920

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,3,x) = Sum_{m=0..n} a(n,m)*x^m have e.g.f. exp(-z*x/(1-z))/(1-z)^4. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} |A008297(n,m)|*(-x)^m, n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).
These polynomials appear in the radial part of the l=1 (p-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
The unsigned version of this triangle is the triangle of unsigned 2-Lah numbers A143497. - Peter Bala, Aug 25 2008
This matrix (unsigned) is embedded in that for n!*L(n,-3,-x). Introduce 0,0,0 to each unsigned row and then add 1,-2,1,4,2,1 to the beginning of the array as the first three rows to generate n!*L(n,-3,-x). - Tom Copeland, Apr 21 2014
From Wolfdieter Lang, Jul 07 2014: (Start)
The standard Rodrigues formula for the monic generalized Laguerre polynomials (also called Laguerre-Sonin polynomials) is Lm(n,alpha,x) := (-1)^n*n!*L(n,3,x) is x^(-alpha)*exp(x)*(d/dx)^n(exp(-x)*x^(n+alpha)).
Another Rodrigues type formula is Lm(n,alpha,x) = exp(x)*x^(-alpha+n+1)*(-x^2*d/dx)^n*(exp(-x)*x^(alpha+1)). This is derivable from the differential - difference relation of Lm(n,alpha,x) which yields first the creation operator formula -(x*d/dx + (-x + alpha + n + 1))*Lm(n,alpha,x) = Lm(n+1,alpha,x) or in the variable q = log(x) the operator -(d/dq + alpha + n + 1 - exp(q)).
The identity (due to Christoph Mayer) (d/dq - (d/dq)W(q))*f(q) = exp(W(q)*d/dq(exp(-W(q)*f(q)) is then iterated with W(q) = W(alpha,n,q) = exp(q) - (alpha + n + 1)*q and a further change of variables leads then to the given result. (End)

			The triangle a(n,m) begins:
n\m       0        1       2     3    4   5 ...
0:        1
1:        4       -1
2:       20      -10      1
3:      120      -90     18     -1
4:      840     -840    252    -28    1
5:     6720    -8400   3360   -560   40  -1
... Formatted by _Wolfdieter Lang_, Jul 07 2014
For more rows see the link.
n = 2: 2!*L(2,3,x) = 20 - 10*x + x^2.

A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.

Indranil Ghosh, Rows 0..125, flattened
Wolfdieter Lang, First eleven rows of the triangle.
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001715, A061206, A062141-A062144. The row sums (signed) give A062146, the row sums (unsigned) give A062147.
Cf. A143497. - Peter Bala, Aug 25 2008
Cf. A062139, A105278. - Wolfdieter Lang, Jul 07 2014

Flatten[Table[((-1)^m)*n!*Binomial[n+3,n-m]/m!,{n,0,9},{m,0,n}]] (* Indranil Ghosh, Feb 23 2017 *)

row(n) = Vecrev(n!*pollaguerre(n, 3)); \\ Michel Marcus, Feb 06 2021

a(n, m) = ((-1)^m)*n!*binomial(n+3, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^4), m >= 0.

A062140 Coefficient triangle of generalized Laguerre polynomials n!*L(n,4,x) (rising powers of x).

Original entry on oeis.org

1, 5, -1, 30, -12, 1, 210, -126, 21, -1, 1680, -1344, 336, -32, 1, 15120, -15120, 5040, -720, 45, -1, 151200, -181440, 75600, -14400, 1350, -60, 1, 1663200, -2328480, 1164240, -277200, 34650, -2310, 77, -1, 19958400, -31933440

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,4,x)= sum(a(n,m)*x^m,m=0..n) have g.f. exp(-z*x/(1-z))/(1-z)^5. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(|A008297(n,m)|*(-x)^m, m=1..n) and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).

			Triangle begins:
  {1};
  {5,-1};
  {30,-12,1};
  {210,-126,21,-1};
  ...
2!*L(2,4,x)=30-12*x+x^2.

Indranil Ghosh, Rows 0..125
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001720(n+4), A062199, A062260-A062263. The row sums (signed) give A062265, the row sums (unsigned) give A062266.

Cf. A021009, A062137-A062139, A066667.

Flatten[Table[((-1)^m)*n!*Binomial[n+4,n-m]/m!,{n,0,11},{m,0,n}]] (* Indranil Ghosh, Feb 23 2017 *)

row(n) = Vecrev(n!*pollaguerre(n, 4)); \\ Michel Marcus, Feb 06 2021

import math
f=math.factorial
def C(n,r):
    return f(n)//f(r)//f(n-r)
i=0
for n in range(26):
    for m in range(n+1):
        print(i, (-1)**m*f(n)*C(n+4,n-m)//f(m))
        i+=1 # Indranil Ghosh, Feb 23 2017

T(n, m) = ((-1)^m)*n!*binomial(n+4, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^5), m >= 0.

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A062193 Fourth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

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A062194 Fifth column sequence of triangle A062139 (generalized a=2 Laguerre).

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A062195 Sixth (unsigned) column sequence of triangle A062139 (generalized a=2 Laguerre).

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A062197 Row sums of signed triangle A062139 (generalized a=2 Laguerre).

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A021009 Triangle of coefficients of Laguerre polynomials n!*L_n(x) (rising powers of x).

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A052852 Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)).

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