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

%I A062138 #32 Mar 22 2021 07:27:11
%S A062138 1,6,-1,42,-14,1,336,-168,24,-1,3024,-2016,432,-36,1,30240,-25200,
%T A062138 7200,-900,50,-1,332640,-332640,118800,-19800,1650,-66,1,3991680,
%U A062138 -4656960,1995840,-415800,46200,-2772,84,-1,51891840,-69189120
%N A062138 Coefficient triangle of generalized Laguerre polynomials n!*L(n,5,x)(rising powers of x).
%C A062138 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).
%C A062138 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.
%C A062138 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.
%C A062138 The unsigned version of this triangle is the triangle of unsigned 3-Lah numbers A143498. - _Peter Bala_, Aug 25 2008
%D A062138 A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.
%H A062138 Indranil Ghosh, <a href="/A062138/b062138.txt">Rows 0..125, flattened</a>
%H A062138 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A062138 T(n, m) = ((-1)^m)*n!*binomial(n+5, n-m)/m!.
%F A062138 E.g.f. for m-th column: ((-x/(1-x))^m)/(m!*(1-x)^6), m >= 0.
%e A062138 Triangle begins:
%e A062138   {1};
%e A062138   {6, -1};
%e A062138   {42, -14, 1};
%e A062138   {336, -168, 24, -1};
%e A062138   ...
%e A062138 2!*L(2, 5, x) = 42-14*x+x^2.
%t A062138 Flatten[Table[((-1)^m)*n!*Binomial[n+5,n-m]/m!,{n,0,8},{m,0,n}]] (* _Indranil Ghosh_, Feb 24 2017 *)
%o A062138 (PARI) 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
%o A062138 (PARI) row(n) = Vecrev(n!*pollaguerre(n, 5)); \\ _Michel Marcus_, Feb 06 2021
%o A062138 (Python)
%o A062138 import math
%o A062138 f=math.factorial
%o A062138 def C(n, r):return f(n)//f(r)//f(n-r)
%o A062138 i=-1
%o A062138 for n in range(26):
%o A062138     for m in range(n+1):
%o A062138         i += 1
%o A062138         print(str(i)+" "+str(((-1)**m)*f(n)*C(n+5, n-m)//f(m))) # _Indranil Ghosh_, Feb 24 2017
%Y A062138 Cf. A021009, A062137, A062139, A062140, A066667.
%Y A062138 For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148, A062149, A062150, A062151, A062152.
%Y A062138 Row sums (signed) give A062191, row sums (unsigned) give A062192.
%Y A062138 Cf. A143498.
%K A062138 sign,easy,tabl
%O A062138 0,2
%A A062138 _Wolfdieter Lang_, Jun 19 2001