A062260 -id:A062260 - cp's OEIS frontend

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

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.

A062261 Fourth (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).

Original entry on oeis.org

1, 32, 720, 14400, 277200, 5322240, 103783680, 2075673600, 42810768000, 913296384000, 20183850086400, 462393656524800, 10981849342464000, 270322445352960000, 6893222356500480000, 181981070211612672000

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. A001720, A062199, A062260.

[Factorial(n+3)*Binomial(n+7,7)/6: n in [0..30]]; // G. C. Greubel, May 13 2018

Table[(n+3)!*Binomial[n+7, 7]/3!, {n, 0, 30}] (* G. C. Greubel, May 13 2018 *)

{ f=2; for (n=0, 100, f*=n + 3; write("b062261.txt", n, " ", f*binomial(n + 7, 7)/6) ) } \\ Harry J. Smith, Aug 03 2009

E.g.f.: (1+21*x+63*x^2+35*x^3)/(1-x)^11.
a(n) = A062140(n+3, 3).
a(n) = (n+3)!*binomial(n+7, 7)/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,-8), (n>=3). - Milan Janjic, Mar 01 2009

A062262 Fifth (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).

Original entry on oeis.org

1, 45, 1350, 34650, 831600, 19459440, 454053600, 10702692000, 256864608000, 6307453152000, 158947819430400, 4118193503424000, 109818493424640000, 3015784780968960000, 85303626661693440000

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. A001720, A062199, A062260, A062261.

[Factorial(n+4)*Binomial(n+8,8)/24: n in [0..30]]; // G. C. Greubel, May 13 2018

Table[(n+4)!*Binomial[n+8, 8]/4!, {n, 0, 30}] (* G. C. Greubel, May 13 2018 *)
With[{nn=20},CoefficientList[Series[(1+32x+168x^2+224x^3+70x^4)/(1-x)^13,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 18 2025 *)

{ f=6; for (n=0, 100, f*=n + 4; write("b062262.txt", n, " ", f*binomial(n + 8, 8)/24) ) } \\ Harry J. Smith, Aug 03 2009

E.g.f.: (1+32*x+168*x^2+224*x^3+70*x^4)/(1-x)^13.
a(n) = A062140(n+4, 4).
a(n) = (n+4)!*binomial(n+8, 8)/4!.
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,-9), (n>=4). - Milan Janjic, Mar 01 2009

A062263 Sixth (unsigned) column of triangle A062140 (generalized a=4 Laguerre).

Original entry on oeis.org

1, 60, 2310, 73920, 2162160, 60540480, 1664863200, 45664819200, 1261490630400, 35321737651200, 1006669523059200, 29284931579904000, 871226714502144000, 26538906072526848000, 828392996692445184000

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. A001720, A062140, A062199, A062260, A062261, A062262.

[Factorial(n+5)*Binomial(n+9, 9)/Factorial(5): n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+5)!*Binomial[n+9,9]/5!, {n, 0, 20}] (* G. c. Greubel, May 12 2018 *)

{ f=24; for (n=0, 100, f*=n + 5; write("b062263.txt", n, " ", f*binomial(n + 9, 9)/120) ) } \\ Harry J. Smith, Aug 03 2009

E.g.f.: N(4;5, x)/(1-x)^15, with N(4;5, x) := Sum_{k=0..5} A062264(5, k)* x^k = 1 + 45*x + 360*x^2 + 840*x^3 + 630*x^4 + 226*x^5.
a(n) = A062140(n+5, 5).
a(n) = (n+5)!*binomial(n+9, 9)/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-9) = (-1)^(n-1)*f(n,9,-6), (n>=9). - Milan Janjic, Mar 01 2009

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

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A062261 Fourth (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).

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A062262 Fifth (unsigned) column sequence of triangle A062140 (generalized a=4 Laguerre).

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A062263 Sixth (unsigned) column of triangle A062140 (generalized a=4 Laguerre).

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