A060338 Triangle T(n,k) of coefficients of Meixner polynomials of degree n, k=0..n.

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, 91, 0, 1519, 0, 3429, 0, 1, 0, 140, 0, 4214, 0, 24940, 0, 11025, 1, 0, 204, 0, 10038, 0, 122156, 0, 230481, 0, 1, 0, 285, 0, 21378, 0, 463490, 0, 2250621, 0, 893025

Offset: 0

Author

Vladeta Jovovic, Mar 30 2001

Keywords

• easy
• nonn
• tabl

Comments

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.

See A060524 for an application to combinatorics. - N. J. A. Sloane, May 30 2013

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]

Examples

			[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, -91, 0, 1519, 0, -3429, 0],
[1, 0, -140, 0, 4214, 0, -24940, 0, 11025],
[1, 0, -204, 0, 10038, 0, -122156, 0, 230481, 0], ...
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,...
		

References

• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Links

• Paul L. Butzer and Tom H. Koornwinder, Josef Meixner: His life and his orthogonal polynomials, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
• A. Hamdi and J. Zeng, Orthogonal polynomials and operator orderings, J. Math. Phys., 51:043506, 2010; arXiv:1006.0808 [math.CO], 2010.
• R. J. Mathar, Gaussian Quadrature of the Integrals Int_(-infty)^infty F(x) dx /cosh(x)
• J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. Lond. Math. Soc. 9 (1934), 6-13.

Crossrefs

Cf. A028353, A060524, A000330 (third column), A214615 (row sums), A214616 (fifth column).

Triangle without zeros: A094368. Unsigned version: A060524.

Programs

Mathematica

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 *)

Maxima

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 */

Formula

E.g.f.: exp(x*arctan(y))/sqrt(1+y^2).