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

%I A094368 #24 Mar 13 2025 12:44:53
%S A094368 1,1,-1,1,-5,1,-14,9,1,-30,89,1,-55,439,-225,1,-91,1519,-3429,1,-140,
%T A094368 4214,-24940,11025,1,-204,10038,-122156,230481,1,-285,21378,-463490,
%U A094368 2250621,-893025,1,-385,41778,-1467290,14466221,-23941125,1,-506
%N A094368 Triangle M(k,n) read by rows: coefficients of Meixner polynomials.
%H A094368 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 A094368 Dominique Foata, <a href="https://www.mat.univie.ac.at/~slc/opapers/s06foata.html">Combinatoire des identités sur les polynomes de Meixner</a>, Sem. Loth. de Comb. B06c (1982).
%H A094368 Nicolas Loizeau, Berislav Buča, and Dries Sels, <a href="https://arxiv.org/abs/2503.07403">Opening Krylov space to access all-time dynamics via dynamical symmetries</a>, arXiv:2503.07403 [quant-ph], 2025. See p. 9.
%H A094368 Josef 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.
%H A094368 M. Micu, <a href="https://doi.org/10.1088/0305-4470/18/16/004">Continuous Hahn polynomials</a>, J. Math. Phys. 34 (3) (1993), 1197-1205.
%H A094368 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MeixnerPolynomialoftheSecondKind.html">Meixner Polynomial of the Second Kind </a>
%F A094368 Recurrence: M(0, z) = 1, M(1, z) = z, M(n+1, z) = z*M(n, z) - n^2*M(n-1, z).
%F A094368 G.f.: exp(z*arctan(x)) / sqrt(1+x^2).
%F A094368 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
%e A094368 z,
%e A094368 z^2 - 1,
%e A094368 z^3 - 5*z,
%e A094368 z^4 - 14*z^2 + 9,
%e A094368 z^5 - 30*z^3 + 89*z,
%e A094368 z^6 - 55*z^4 + 439*z^2 - 225,
%e A094368 z^7 - 91*z^5 + 1519*z^3 - 3429*z,
%e A094368 z^8 - 140*z^6 + 4214*z^4 - 24940*z^2 + 11025,
%e A094368 z^9 - 204*z^7 + 10038*z^5 - 122156*z^3 + 230481*z,
%Y A094368 Essentially the same as A060338.
%Y A094368 Cf. A060524.
%K A094368 sign,tabf
%O A094368 1,5
%A A094368 _Ralf Stephan_, Jun 03 2004