A094368 Triangle M(k,n) read by rows: coefficients of Meixner polynomials.

1, 1, -1, 1, -5, 1, -14, 9, 1, -30, 89, 1, -55, 439, -225, 1, -91, 1519, -3429, 1, -140, 4214, -24940, 11025, 1, -204, 10038, -122156, 230481, 1, -285, 21378, -463490, 2250621, -893025, 1, -385, 41778, -1467290, 14466221, -23941125, 1, -506

Offset

1,5

Links

Table of n, a(n) for n=1..43.

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.

D. Foata, Combinatoire des identités sur les polynomes de Meixner, Sem. Loth. de Comb. B06c (1982).

J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. Lond. Math. Soc. 9 (1934), 6-13.

M. Micu, Continuous Hahn polynomials, J. Math. Phys. 34 (3) (1993), 1197-1205.

Eric Weisstein's World of Mathematics, Meixner Polynomial of the Second Kind

Formula

Recurrence: M(0, z) = 1, M(1, z) = z, M(n+1, z) = z*M(n, z) - n^2*M(n-1, z).

G.f.: exp(z*arctan(x)) / sqrt(1+x^2).

The n-th (unsigned) row polynomial R(n, x) = (-i)^n * M(n, i*x) = n!*Sum_{k = 0..n} 2^k*binomial(n, k)*binomial(x/2 - 1/2, k). - Peter Bala, Mar 10 2024

Example

z,
z^2 - 1,
z^3 - 5*z,
z^4 - 14*z^2 + 9,
z^5 - 30*z^3 + 89*z,
z^6 - 55*z^4 + 439*z^2 - 225,
z^7 - 91*z^5 + 1519*z^3 - 3429*z,
z^8 - 140*z^6 + 4214*z^4 - 24940*z^2 + 11025,
z^9 - 204*z^7 + 10038*z^5 - 122156*z^3 + 230481*z,

Crossrefs

Essentially the same as A060338.

Cf. A060524.

Sequence in context: A104792 A120393 A370518 * A295574 A087727 A039807

Adjacent sequences: A094365 A094366 A094367 * A094369 A094370 A094371

Keyword

sign,tabf

Author

Ralf Stephan, Jun 03 2004

Status

approved