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%I #49 Feb 06 2021 07:08:49
%S 1,4,-1,20,-10,1,120,-90,18,-1,840,-840,252,-28,1,6720,-8400,3360,
%T -560,40,-1,60480,-90720,45360,-10080,1080,-54,1,604800,-1058400,
%U 635040,-176400,25200,-1890,70,-1,6652800,-13305600,9313920
%N Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x).
%C 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).
%C 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.
%C The unsigned version of this triangle is the triangle of unsigned 2-Lah numbers A143497. - _Peter Bala_, Aug 25 2008
%C 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
%C From _Wolfdieter Lang_, Jul 07 2014: (Start)
%C 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)).
%C 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)).
%C 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)
%D A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.
%H Indranil Ghosh, <a href="/A062137/b062137.txt">Rows 0..125, flattened</a>
%H Wolfdieter Lang, <a href="/A062137/a062137.pdf">First eleven rows of the triangle.</a>
%H <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F a(n, m) = ((-1)^m)*n!*binomial(n+3, n-m)/m!.
%F E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^4), m >= 0.
%e The triangle a(n,m) begins:
%e n\m 0 1 2 3 4 5 ...
%e 0: 1
%e 1: 4 -1
%e 2: 20 -10 1
%e 3: 120 -90 18 -1
%e 4: 840 -840 252 -28 1
%e 5: 6720 -8400 3360 -560 40 -1
%e ... Formatted by _Wolfdieter Lang_, Jul 07 2014
%e For more rows see the link.
%e n = 2: 2!*L(2,3,x) = 20 - 10*x + x^2.
%t Flatten[Table[((-1)^m)*n!*Binomial[n+3,n-m]/m!,{n,0,9},{m,0,n}]] (* _Indranil Ghosh_, Feb 23 2017 *)
%o (PARI) row(n) = Vecrev(n!*pollaguerre(n, 3)); \\ _Michel Marcus_, Feb 06 2021
%Y For m=0..5 the (unsigned) columns give A001715, A061206, A062141-A062144. The row sums (signed) give A062146, the row sums (unsigned) give A062147.
%Y Cf. A143497. - _Peter Bala_, Aug 25 2008
%Y Cf. A062139, A105278. - _Wolfdieter Lang_, Jul 07 2014
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_, Jun 19 2001
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