cp's OEIS Frontend

A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!*2^n*Lag(n,x/2,1/2) of order 1/2.

%I A130757 #50 Jan 19 2022 22:39:11
%S A130757 1,3,-1,15,-10,1,105,-105,21,-1,945,-1260,378,-36,1,10395,-17325,6930,
%T A130757 -990,55,-1,135135,-270270,135135,-25740,2145,-78,1,2027025,-4729725,
%U A130757 2837835,-675675,75075,-4095,105,-1,34459425,-91891800,64324260,-18378360,2552550,-185640,7140,-136
%N A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!*2^n*Lag(n,x/2,1/2) of order 1/2.
%C A130757 These polynomials appear in the radial l=0 (s) wave functions of the isotropic three-dimensional harmonic quantum oscillator with the dimensionless variable x=(r/L)^2 with r>=0 and L^2=h/(m*f0). h is Planck's constant and m and f0 are the mass and the frequency of the oscillator.
%C A130757 From _Tom Copeland_, Dec 13 2015: (Start)
%C A130757 See A099174 for relations to the Hermite polynomials and the link in A176230 for operator relations. The infinitesimal generator for this matrix contains A014105.
%C A130757 The row polynomials are P(n,x) = 2^n n! Lag(n,x/2,1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q, with raising operator R = -x^(-2) [x^(3/2) (1 - 2D)]^2 = 3 - x + (4x - 6) D - 4x D^2 with D = d/dx, i.e., R P(n,x) - P(n+1,x). A matrix reresentation of R acting on an o.g.f. (formal power series) is given by the transpose of the production matrix below. The diagonal corresponds to (3 + 4 xD) x^n = (3 + 4n) x^n; the upper diagonal, to -x x^n = -x^(n+1); and the lower diagonal, to (-6 - 4 xD) D x^n = -n (6 + 4(n-1)) x^(n-1), the sequence A002943. See A176230 for a similar relation.
%C A130757 The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
%C A130757 (End)
%C A130757 Exponential Riordan array [1/(1-2x)^(3/2), -x/(1-2x)]. - _Paul Barry_, Mar 07 2017
%H A130757 Robert Israel, <a href="/A130757/b130757.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 to 140, flattened)
%H A130757 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A130757 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 775, 22.3.9.
%H A130757 T. Copeland, <a href="https://tcjpn.wordpress.com/2020/07/15/juggling-zeros-in-the-matrix-example-ii/">Juggling Zeros in the Matrix: Example II</a>, 2020.
%H A130757 Wolfdieter Lang, <a href="/A130757/a130757.txt">First ten rows and more</a>.
%F A130757 a(n,m) = n!*(2^(n-m))*L(1/2,n,m) with L(1/2,n,m) = ((-1)^m)*binomial(n+1/2,n-m)/m!, n >= m >= 0, otherwise 0.
%F A130757 Let IP be the lower triangular matrix with its first subdiagonal equal to the first subdiagonal (cf. A014105) of this entry's unsigned matrix M and with all other elements equal to zero. Then IP is the infinitesimal generator of M, i.e., M = exp(IP). - _Tom Copeland_, Dec 12 2015
%F A130757 From _Tom Copeland_, Dec 14 2015: (Start)
%F A130757 Production matrix is
%F A130757    3,   -1;
%F A130757   -6,    7,   -1;
%F A130757    0,  -20,   11,   -1;
%F A130757    0,    0,  -42,   15,   -1;
%F A130757    0,    0,    0,  -72,   19,   -1;
%F A130757    0,    0,    0,    0, -110,   23,   -1;
%F A130757    0,    0,    0,    0,    0, -156,   27,   -1;
%F A130757    0,    0,    0,    0,    0,    0, -210,   31,   -1;
%F A130757    0,    0,    0,    0,    0,    0,    0, -272,   35,   -1;
%F A130757   ... (End)
%e A130757 [1]; [3,-1]; [15,-10,1]; [105,-105,21,-1]; [945,-1260,378,-36,1]; ...
%p A130757 seq(seq(n!*2^(n-m)*(-1)^m*binomial(n+1/2,n-m)/m!,m=0..n),n=0..10); # _Robert Israel_, Dec 25 2015
%t A130757 Table[n! (2^(n - m)) ((-1)^m) Binomial[n + 1/2, n - m]/m!, {n, 0, 8}, {m, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 24 2015 *)
%Y A130757 Cf. A021009 (Coefficient table of n!*L(n, 0, x)).
%Y A130757 Row sums (signed) give A131441. Row sums (unsigned) give A066224.
%Y A130757 Cf. A001147, A002943, A014105, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299, A176230.
%K A130757 sign,tabl,easy
%O A130757 0,2
%A A130757 _Wolfdieter Lang_, Jul 13 2007
%E A130757 Title formula corrected by _Tom Copeland_, Dec 12 2015