A062137 -id:A062137 - cp's OEIS frontend

A062147 Row sums of unsigned triangle A062137 (generalized a=3 Laguerre).

1, 5, 31, 229, 1961, 19081, 207775, 2501801, 32989969, 472630861, 7307593151, 121247816845, 2148321709561, 40476722545169, 807927483311551, 17028146983530961, 377844723929464865, 8803698102396787861, 214877019857456672479, 5482159931449737760181

Offset: 0

Wolfdieter Lang, Jun 19 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Index entries for sequences related to Laguerre polynomials

Cf. A062137, A216294.

[Factorial(n)*(&+[Binomial(n+3,n-m)/Factorial(m): m in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 06 2018

A062147 := n -> n!*simplify(LaguerreL(n,3,-1), 'LaguerreL');
seq(A062147(n), n = 0 .. 30); # G. C. Greubel, Mar 10 2021

Table[Sum[n!*Binomial[n+3,n-k]/k!,{k,0,n}],{n,0,20}]
(* or *)
Table[n!*SeriesCoefficient[E^(x/(1-x))/(1-x)^4,{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 11 2012 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(x/(1-x))/(1-x)^4)) \\ Joerg Arndt, May 06 2013

a(n) = vecsum(apply(abs,Vec(n!*pollaguerre(n, 3)))); \\ Michel Marcus, Feb 06 2021

[factorial(n)*gen_laguerre(n, 3, -1) for n in (0..30)] # G. C. Greubel, Mar 10 2021

E.g.f.: exp(x/(1-x))/(1-x)^4.
a(n) = Sum_{m=0..n} n!*binomial(n+3, n-m)/m!.
a(n) = (2*n+3)*a(n-1) - (n-1)*(n+2)*a(n-2). - Vaclav Kotesovec, Oct 11 2012
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+7/4)/sqrt(2). - Vaclav Kotesovec, Oct 11 2012
a(n) = n!*LaguerreL(n, 3, -1). - G. C. Greubel, Mar 10 2021

A062141 Third column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

Original entry on oeis.org

1, 18, 252, 3360, 45360, 635040, 9313920, 143700480, 2335132800, 39956716800, 719220902400, 13599813427200, 269729632972800, 5602076992512000, 121645100408832000, 2757288942600192000, 65140951268929536000, 1601701037083090944000, 40932359836567879680000

Offset: 0

Wolfdieter Lang, Jun 19 2001

			a(2) = (2+2)! * binomial(2+5,5) / 2! = (24*21)/2 = 252. - _Indranil Ghosh_, Feb 24 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Index entries for sequences related to Laguerre polynomials

Cf. A061206, A062137.

[Factorial(n+2)*Binomial(n+5,5)/2: n in [0..20]]; // G. C. Greubel, May 11 2018

a:= n->(n+2)!*binomial(n+5, 5)/2!: seq(a(n), n=0..20); # Zerinvary Lajos, Apr 29 2007

Table[(n+2)!*Binomial[n+5,5]/2!,{n,0,15}] (* Indranil Ghosh, Feb 24 2017 *)

a(n)=(n+2)!*binomial(n+5,5)/2! \\ Indranil Ghosh, Feb 24 2017

x='x+O('x^30); Vec(serlaplace((1+10*x+10*x^2)/(1-x)^8)) \\ G. C. Greubel, May 11 2018

import math
f=math.factorial
def C(n,r):return f(n)/f(r)/f(n-r)
def A062141(n): return f(n+2)*C(n+5,5)/f(2) # Indranil Ghosh, Feb 24 2017

[binomial(n,5)*factorial (n-3)/2 for n in range(5, 21)] # Zerinvary Lajos, Jul 07 2009

a(n) = (n+2)!*binomial(n+5, 5)/2!.
E.g.f.: (1 + 10*x + 10*x^2)/(1-x)^8.
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,-3), (n >= 5). - Milan Janjic, Mar 01 2009

A062144 Sixth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

Original entry on oeis.org

1, 54, 1890, 55440, 1496880, 38918880, 998917920, 25686460800, 667847980800, 17660868825600, 476843458291200, 13178219210956800, 373382877643776000, 10856825211488256000, 324153781314435072000

Offset: 0

Wolfdieter Lang, Jun 19 2001

			a(2) = (2+5)! * binomial(2+8,8)/ 5! = (5040 * 45) / 120 = 1890. - _Indranil Ghosh_, Feb 24 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Index entries for sequences related to Laguerre polynomials

Cf. A062143.

[Factorial(n+5)*Binomial(n+8, 8)/Factorial(5): n in [0..20]]; // G. C. Greubel, May 11 2018

Table[(n+5)!*Binomial[n+8,8]/5!,{n,0,14}] (* Indranil Ghosh, Feb 24 2017 *)

a(n)=(n+5)!*binomial(n+8, 8)/5! \\ Indranil Ghosh, Feb 24 2017

import math
f=math.factorial
def C(n, r):return f(n)/f(r)/f(n-r)
def A062144(n): return f(n+5)*C(n+8, 8)/f(5) # Indranil Ghosh, Feb 24 2017

a(n) = (n+5)!*binomial(n+8, 8)/5!.
E.g.f.: N(3;5, x)/(1-x)^14 with N(3;5, x) := Sum_{k=0..5} A062145(5, k) *x^k = 1 +40*x +280*x^2 +560*x^3 +350*x^4 +56*x^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, -9), (n>=5). - Milan Janjic, Mar 01 2009

A062146 Row sums of signed triangle A062137 (generalized Laguerre, a=3).

Original entry on oeis.org

1, 3, 11, 47, 225, 1159, 6067, 28419, 58433, -1390645, -35514309, -636045257, -10431927839, -167173905393, -2678202265885, -43236880758901, -703702453254783, -11485574113211501, -185707408082199317, -2901041900411825985

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. A062137.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(-x/(1-x))/(1-x)^4)); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 11 2018

Table[n!*LaguerreL[n, 3, 1],{n,0,20}] (* Vaclav Kotesovec, Aug 01 2013 *)

{ f=1; for (n=0, 100, if (n>1, f*=n); a=f*binomial(n+3, n); g=1; a+=sum(m=1, n, ((-1)^m)*f*binomial(n+3, n-m)/g*=m); write("b062146.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 02 2009

my(x='x+O('x^30)); Vec(exp(-x/(1-x))/(1-x)^4) \\ G. C. Greubel, May 11 2018

a(n) = vecsum(Vec(n!*pollaguerre(n, 3))); \\ Michel Marcus, Feb 06 2021

E.g.f.: exp(-x/(1-x))/(1-x)^4.
a(n) = sum(((-1)^m)*n!*binomial(n+3, n-m)/m!, m=0..n).
a(n) = (2*n+1)*a(n-1) - (n-1)*(n+2)*a(n-2). - Vaclav Kotesovec, Aug 01 2013

A062142 Fourth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

Original entry on oeis.org

1, 28, 560, 10080, 176400, 3104640, 55883520, 1037836800, 19978358400, 399567168000, 8310997094400, 179819755315200, 4045944494592000, 94612855873536000, 2297740785500160000, 57903067794604032000

Offset: 0

Wolfdieter Lang, Jun 19 2001

			a(3) = (3+3)!*binomial(3+6,6)/3! = (720*84)/6 = 10080. - _Indranil Ghosh_, Feb 23 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Index entries for sequences related to Laguerre polynomials

Cf. A062137, A062141.

[Factorial(n+3)*Binomial(n+6,6)/6: n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+3)!*Binomial[n+6,6]/3!,{n,0,15}] (* Indranil Ghosh, Feb 23 2017 *)

a(n) =(n+3)!*binomial(n+6,6)/3! \\ Indranil Ghosh, Feb 23 2017

import math
f=math.factorial
def C(n,r):
    return f(n)/f(r)/f(n-r)
def A062142(n):return f(n+3)*C(n+6,6)/f(3) # Indranil Ghosh, Feb 23 2017

[binomial(n,6)*factorial(n-3)/factorial(3) for n in range(6, 22)] # Zerinvary Lajos, Jul 07 2009

a(n) = (n+3)!*binomial(n+6, 6)/3!; e.g.f.: (1 + 18*x + 45*x^2 + 20*x^3)/(1-x)^10.

If we define f(n,i,x) = Sum_{k=1..n} Sum_{j=1..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j), then a(n-3) = (-1)^(n-1)*f(n,3,-7), (n>=3). - Milan Janjic, Mar 01 2009

A062143 Fifth column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

Original entry on oeis.org

1, 40, 1080, 25200, 554400, 11975040, 259459200, 5708102400, 128432304000, 2968213248000, 70643475302400, 1733976211968000, 43927397369856000, 1148870392750080000, 31019500604252160000, 864410083505160192000

Offset: 0

Wolfdieter Lang, Jun 19 2001

The coefficients of the numerator polynomials N(m,x) of the e.g.f. for column m (here m=4) give triangle A062145.

			a(3) = (3+4)! * binomial(3+7,7) / 4! = (5040 * 120) / 24 = 25200. - _Indranil Ghosh_, Feb 23 2017

Indranil Ghosh, Table of n, a(n) for n = 0..400
Index entries for sequences related to Laguerre polynomials

Cf. A062137, A062142, A062145.

[Factorial(n+4)*Binomial(n+7,7)/Factorial(4): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+4)!*Binomial[n+7,7]/4!,{n,0,15}] (* Indranil Ghosh, Feb 23 2017 *)

a(n) = (n+4)!*binomial(n+7,7)/4! \\ Indranil Ghosh, Feb 23 2017

import math
f=math.factorial
def C(n,r):return f(n)/f(r)/f(n-r)
def A062143(n):return f(n+4)*C(n+7,7)/f(4) # Indranil Ghosh, Feb 23 2017

a(n) = (n+4)!*binomial(n+7, 7)/4!;
E.g.f.: (1 + 28*x + 126*x^2 + 140*x^3 + 35*x^4)/(1-x)^12.
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-4) = (-1)^n*f(n,4,-8), (n>=4). - Milan Janjic, Mar 01 2009

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

A062145 Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.

Original entry on oeis.org

1, 1, 4, 1, 10, 10, 1, 18, 45, 20, 1, 28, 126, 140, 35, 1, 40, 280, 560, 350, 56, 1, 54, 540, 1680, 1890, 756, 84, 1, 70, 945, 4200, 7350, 5292, 1470, 120, 1, 88, 1540, 9240, 23100, 25872, 12936, 2640, 165, 1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220

Offset: 0

Wolfdieter Lang, Jun 19 2001

Coefficient triangle of certain polynomials N(3; m,x).

			As a square array:
    1,    1,     1,     1,     1,     1,    1,  1, ... A000012;
    4,   10,    18,    28,    40,    54,   70, 88, ... A028552;
   10,   45,   126,   280,   540,   945, 1540, ....... A105938;
   20,  140,   560,  1680,  4200,  9240, ............. A105939;
   35,  350,  1890,  7350, 23100, 62370, ............. A027803;
   56,  756,  5292, 25872, 99792, .................... A105940;
   84, 1470, 12936, 77616, ........................... A105942;
  120, 2640, 28512, .................................. A105943;
  165, 4455, 57015, .................................. A105944;
  ....;
As a triangle:
  1;
  1,   4;
  1,  10,   10;
  1,  18,   45,    20;
  1,  28,  126,   140,    35;
  1,  40,  280,   560,   350,    56;
  1,  54,  540,  1680,  1890,   756,    84;
  1,  70,  945,  4200,  7350,  5292,  1470,   120;
  1,  88, 1540,  9240, 23100, 25872, 12936,  2640,  165;
  1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220;
  ....;

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

Family of polynomials: A008459 (c=1), A132813 (c=2), A062196 (c=3), this sequence (c=4), A062264 (c=5), A062190 (c=6).
Columns: A028552 (k=1), A105938 (k=2), A105939 (k=3), A027803 (k=4), A105940 (k=5), A105942 (k=6), A105943 (k=7), A105944 (k=8).
Diagonals: A000292 (k=n), A027800 (k=n-1), A107417 (k=n-2), A107418 (k=n-3), A107419 (k=n-4), A107420 (k=n-5), A107421 (k=n-6), A107422 (k=n-7).
Sums: A002054 (row).
Cf. A000108, A000292, A047074.

A062145:= func< n,k | Binomial(n,k)*Binomial(n+3,k) >;
[A062145(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 07 2025

NN[3, m_, x_] := x^m*(2*m+3)!*Hypergeometric2F1[-m, -m, -2*m-3, (x-1)/x]/( (m+3)!*m!); Table[CoefficientList[NN[3, m, x], x], {m, 0, 9}] // Flatten (* Jean-François Alcover, Sep 18 2013 *)
P[c_, n_, z_] := Sum[Binomial[n, k] Pochhammer[n-k+c, k] z^k /k!, {k,0,n}];
CL[c_] := Table[CoefficientList[P[c, n, z], z], {n, 0, 5}] // TableForm
CL[4]  (* Peter Luschny, Feb 12 2024 *)
A062145[n_,k_]:= Binomial[n,k]*Binomial[n+3,k];
Table[A062145[n,k], {n,0,12},{k,0,n}]//Flatten (* G. C. Greubel, Mar 07 2025 *)

def A062145(n,k): return binomial(n,k)*binomial(n+3,k)
print(flatten([[A062145(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 07 2025

The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=3) Laguerre triangle L(3; n+m, m) = A062137(n+m, m), n >= 0, is N(3; m, x)/(1-x)^(2*(m+2)), with the row polynomials N(3; m, x) := Sum_{k=0..m} a(m, k)*x^k.
N(3; m, x) := ((1-x)^(2*(m+2)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+4))); a(m, k) = [x^k]N(3; m, x).
N(3; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+3-j)!/((m+3)!*(m-j)!))*(x^(m-j))*(1-x)^j).
N(3; m, x)= x^m*(2*m+3)! * 2F1(-m, -m; -2*m-3; (x-1)/x)/((m+3)!*m!). - Jean-François Alcover, Sep 18 2013
From G. C. Greubel, Mar 07 2025 : (Start)
T(n, k) = binomial(n, k)*binomial(n+3, k).
T(2*n, n) = (1/2)*(n+1)^2*A000108(n)*A000108(n+2).
Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^floor((n+2)/2)*(A047074(n+3) - A047074(n+ 2)). (End)

New name by Peter Luschny, Feb 12 2024

More terms from G. C. Greubel, Mar 07 2025

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)

A062139 Coefficient triangle of generalized Laguerre polynomials n!*L(n,2,x) (rising powers of x).

Original entry on oeis.org

1, 3, -1, 12, -8, 1, 60, -60, 15, -1, 360, -480, 180, -24, 1, 2520, -4200, 2100, -420, 35, -1, 20160, -40320, 25200, -6720, 840, -48, 1, 181440, -423360, 317520, -105840, 17640, -1512, 63, -1, 1814400, -4838400, 4233600

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,2,x) = Sum_{m=0..n} a(n,m)*x^m have e.g.f. exp(-z*x/(1-z))/(1-z)^3. 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).
This unsigned matrix is embedded in the matrix for n!*L(n,-2,-x). Introduce 0,0 to each unsigned row and then add 1,-1,1 to the array as the first two rows to generate n!*L(n,-2,-x). - Tom Copeland, Apr 20 2014
The unsigned n-th row reverse polynomial equals the numerator polynomial of the finite continued fraction 1 - x/(1 + (n+1)*x/(1 + n*x/(1 + n*x/(1 + ... + 2*x/(1 + 2*x/(1 + x/(1 + x/(1)))))))). Cf. A089231. The denominator polynomial of the continued fraction is the (n+1)-th row polynomial of A144084. An example is given below. - Peter Bala, Oct 06 2019

			Triangle begins:
     1;
     3,    -1;
    12,    -8,    1;
    60,   -60,   15,   -1;
   360,  -480,  180,  -24,  1;
  2520, -4200, 2100, -420, 35, -1;
  ...
2!*L(2,2,x) = 12 - 8*x + x^2.
Unsigned row 3 polynomial in reverse form as the numerator of a continued fraction: 1 - x/(1 + 4*x/(1 + 3*x/(1 + 3*x/(1 + 2*x/(1 + 2*x/(1 + x/(1 + x))))))) = (60*x^3 + 60*x^2 + 15*x + 1)/(24*x^4 + 96*x^3 + 72*x^2 + 16*x + 1). - _Peter Bala_, Oct 06 2019

Indranil Ghosh, Rows 0..125, flattened
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001710, A005990, A005461, A062193-A062195. The row sums (signed) give A062197, the row sums (unsigned) give A052852.

Cf. A021009, A062137-A062140, A066667, A089231, A144084.

with(PolynomialTools):
p := n -> (n+2)!*hypergeom([-n],[3],x)/2:
seq(CoefficientList(simplify(p(n)), x), n=0..9); # Peter Luschny, Apr 08 2015

Flatten[Table[((-1)^m)*n!*Binomial[n+2,n-m]/m!,{n,0,8},{m,0,n}]] (* Indranil Ghosh, Feb 24 2017 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(((-1)^k)*n!*binomial(n+2, n-k)/k!, ", ");); print(););} \\ Michel Marcus, May 06 2014

row(n) = Vecrev(n!*pollaguerre(n, 2)); \\ 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(16):
    for m in range(n+1):
        i += 1
        print(i,((-1)**m)*f(n)*C(n+2,n-m)//f(m)) # Indranil Ghosh, Feb 24 2017

from functools import cache
@cache
def T(n, k):
    if k < 0 or k > n: return 0
    if k == n: return (-1)**n
    return (n + k + 2) * T(n-1, k) - T(n-1, k-1)
for n in range(7): print([T(n,k) for k in range(n + 1)])
# Peter Luschny, Mar 25 2024

T(n, m) = ((-1)^m)*n!*binomial(n+2, n-m)/m!.
E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^3), m >= 0.
n!*L(n,2,x) = (n+2)!*hypergeom([-n],[3],x)/2. - Peter Luschny, Apr 08 2015
From Werner Schulte, Mar 24 2024: (Start)
T(n, k) = (n+k+2) * T(n-1, k) - T(n-1, k-1) with initial values T(0, 0) = 1 and T(i, j) = 0 if j < 0 or j > i.
T = T^(-1), i.e., T is matrix inverse of T. (End)

cp's OEIS Frontend

A062147 Row sums of unsigned triangle A062137 (generalized a=3 Laguerre).

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A062141 Third column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

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A062144 Sixth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

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A062146 Row sums of signed triangle A062137 (generalized Laguerre, a=3).

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A062142 Fourth (unsigned) column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

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A062143 Fifth column sequence of coefficient triangle A062137 of generalized Laguerre polynomials n!*L(n,3,x).

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