|
%I #40 Mar 18 2024 11:47:37
%S 1,1,0,1,0,1,1,0,5,0,1,0,14,0,9,1,0,30,0,89,0,1,0,55,0,439,0,225,1,0,
%T 91,0,1519,0,3429,0,1,0,140,0,4214,0,24940,0,11025,1,0,204,0,10038,0,
%U 122156,0,230481,0,1,0,285,0,21378,0,463490,0,2250621,0,893025
%N Triangle T(n,k) of coefficients of Meixner polynomials of degree n, k=0..n.
%C The Meixner polynomials M_n(x) satisfy the recurrence: M_(k+1)=x*M_k-k^2*M_(k-1), M_(-1)=0, M_0=1.
%C See A060524 for an application to combinatorics. - _N. J. A. Sloane_, May 30 2013
%C The Meixner polynomials M_n(x) satisfy: M_n(x)=n!*sum(m=0..n/2, binomial(2*m,m)*sum(j=m..n/2, (-1)^(j)*x^(n-2*j)*sum(i=0..2*j-2*m, (2^(i-2*m)*stirling1(i+n+(-2)*j,n-2*j)*binomial(n-2*m-1,2*j-2*m-i))/(i+n+(-2)*j)!))). [_Vladimir Kruchinin_, Sep 25 2013]
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%H Paul L. Butzer and Tom H. Koornwinder, <a href="https://doi.org/10.1016/j.indag.2018.09.009">Josef Meixner: His life and his orthogonal polynomials</a>, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
%H A. Hamdi and J. Zeng, <a href="http://arxiv.org/abs/1006.0808">Orthogonal polynomials and operator orderings</a>, J. Math. Phys., 51:043506, 2010; arXiv:1006.0808 [math.CO], 2010.
%H R. J. Mathar, <a href="http://vixra.org/abs/1303.0038">Gaussian Quadrature of the Integrals Int_(-infty)^infty F(x) dx /cosh(x)</a>
%H J. Meixner, <a href="https://doi.org/10.1112/jlms/s1-9.1.6">Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion</a>, J. Lond. Math. Soc. 9 (1934), 6-13.
%F E.g.f.: exp(x*arctan(y))/sqrt(1+y^2).
%e [1],
%e [1, 0],
%e [1, 0, -1],
%e [1, 0, -5, 0],
%e [1, 0, -14, 0, 9],
%e [1, 0, -30, 0, 89, 0],
%e [1, 0, -55, 0, 439, 0, -225],
%e [1, 0, -91, 0, 1519, 0, -3429, 0],
%e [1, 0, -140, 0, 4214, 0, -24940, 0, 11025],
%e [1, 0, -204, 0, 10038, 0, -122156, 0, 230481, 0], ...
%e M_1(x)=x, M_2(x)=x^2-1, M_3(x)=x^3-5*x, M_4(x)=x^4-14*x^2+9, M_5(x)=x^5-30*x^3+89*x, M_6(x)=x^6-55*x^4+439*x^2-225,...
%t m[0] = 1; m[1] = x; m[k_] := m[k] = x*m[k - 1] - (k - 1)^2*m[k - 2]; row[n_] := CoefficientList[m[n], x] // Reverse // Abs; Table[row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Mar 26 2013 *)
%o (Maxima)
%o M(n,x):=n!*sum(binomial(2*m,m)*sum(((sum((2^(i-2*m)*stirling1(i+n-2*j,n-2*j)*binomial(n-2*m-1,2*j-2*m-i))/(i+n-2*j)!,i,0,2*j-2*m))*(-1)^(j)*x^(n-2*j)),j,m,n/2),m,0,n/2); [_Vladimir Kruchinin_, Sep 25 2013]
%Y Cf. A028353, A060524, A000330 (third column), A214615 (row sums), A214616 (fifth column).
%Y Triangle without zeros: A094368. Unsigned version: A060524.
%K easy,nonn,tabl
%O 0,9
%A _Vladeta Jovovic_, Mar 30 2001
|
