A062149 -id:A062149 - cp's OEIS frontend

A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).

1, 6, -1, 42, -14, 1, 336, -168, 24, -1, 3024, -2016, 432, -36, 1, 30240, -25200, 7200, -900, 50, -1, 332640, -332640, 118800, -19800, 1650, -66, 1, 3991680, -4656960, 1995840, -415800, 46200, -2772, 84, -1, 51891840, -69189120

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,5,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^6. 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 sum(|A008297(n,m)|*(-x)^m, m=1..n), 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=2 (d-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148-A062152. Row sums (signed) give A062191; row sums (unsigned) give A062192.
The unsigned version of this triangle is the triangle of unsigned 3-Lah numbers A143498. - Peter Bala, Aug 25 2008

			Triangle begins:
  {1};
  {6, -1};
  {42, -14, 1};
  {336, -168, 24, -1};
  ...
2!*L(2, 5, x) = 42-14*x+x^2.

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

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

Cf. A021009, A062137, A062139, A062140, A066667.
For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148, A062149, A062150, A062151, A062152.
Row sums (signed) give A062191, row sums (unsigned) give A062192.
Cf. A143498.

Flatten[Table[((-1)^m)*n!*Binomial[n+5,n-m]/m!,{n,0,8},{m,0,n}]] (* Indranil Ghosh, Feb 24 2017 *)

tabl(nn) = {for (n=0, nn, for (m=0, n, print1(((-1)^m)*n!*binomial(n+5, n-m)/m!, ", "); ); print(); ); } \\ Indranil Ghosh, Feb 24 2017

row(n) = Vecrev(n!*pollaguerre(n, 5)); \\ Michel Marcus, Feb 06 2021

import math
f=math.factorial
def C(n, r):return f(n)//f(r)//f(n-r)
i=-1
for n in range(26):
    for m in range(n+1):
        i += 1
        print(str(i)+" "+str(((-1)**m)*f(n)*C(n+5, n-m)//f(m))) # Indranil Ghosh, Feb 24 2017

T(n, m) = ((-1)^m)*n!*binomial(n+5, n-m)/m!.

E.g.f. for m-th column: ((-x/(1-x))^m)/(m!*(1-x)^6), m >= 0.

A062150 Fourth (unsigned) column sequence of triangle A062138 (generalized a=5 Laguerre).

Original entry on oeis.org

1, 36, 900, 19800, 415800, 8648640, 181621440, 3891888000, 85621536000, 1940754816000, 45413662694400, 1098184934246400, 27454623356160000, 709596419051520000, 18956361480376320000, 523195576858386432000

Offset: 0

Wolfdieter Lang, Jun 19 2001

			a(2) = (2+3)! * binomial(2+8,8) / 3! = (120 * 45) / 6 = 900. - _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. A062149.

[Factorial(n+3)*Binomial(n+8, 8)/6: n in [0..20]]; // G. C. Greubel, May 12 2018

Table[(n+3)!*Binomial[n+8,8]/3!,{n,0,15}] (* Indranil Ghosh, Feb 24 2017 *)

a(n)=(n+3)!*binomial(n+8, 8)/3! \\ Indranil Ghosh, Feb 24 2017

import math
f=math.factorial
def C(n, r):return f(n)/f(r)/f(n-r)
def A062150(n): return f(n+3)*C(n+8, 8)/f(3) # Indranil Ghosh, Feb 24 2017

E.g.f.: (1+24*x+84*x^2+56*x^3)/(1-x)^12.
a(n) = A062138(n+3, 3).
a(n) = (n+3)!*binomial(n+8, 8)/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,-9), (n>=3). - Milan Janjic, Mar 01 2009

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A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).

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A062150 Fourth (unsigned) column sequence of triangle A062138 (generalized a=5 Laguerre).

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