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A369117 Table read by rows. T(n, k) = [z^k] LommelR(n, n, 1/z) where LommelR are the Lommel polynomials.

%I A369117 #23 Feb 16 2025 08:34:06
%S A369117 1,0,2,-1,0,24,0,-16,0,480,1,0,-360,0,13440,0,42,0,-10752,0,483840,-1,
%T A369117 0,1728,0,-403200,0,21288960,0,-80,0,79200,0,-18247680,0,1107025920,1,
%U A369117 0,-5280,0,4118400,0,-968647680,0,66421555200,0,130,0,-349440,0,242161920,0,-59041382400,0,4516665753600
%N A369117 Table read by rows. T(n, k) = [z^k] LommelR(n, n, 1/z) where LommelR are the Lommel polynomials.
%C A369117 Lommel polynomials R(n, v, z) are rational functions and not polynomials. We consider here the modified Lommel polynomials h(n, v, z) = R(n, v, 1/z) in the case v = n.
%D A369117 Eugen von Lommel, Zur Theorie der Bessel'schen Functionen, Math. Ann. 4, 103-116 (1871).
%H A369117 David Dickinson, <a href="https://www.ams.org/journals/proc/1954-005-06/S0002-9939-1954-0086897-8/S0002-9939-1954-0086897-8.pdf">On Lommel and Bessel polynomials</a>, AMS Proceedings 1954.
%H A369117 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LommelPolynomial.html">Lommel Polynomial</a>.
%F A369117 T(n, k) = (-1)^binomial(n - k, 2) * (2*z)^n * [z^k] ((Gamma(2*n)/Gamma(n)) * hypergeom([(1-n)/2, -n/2], [n, -n, 1 - 2*n], z^(-2))) for n > 0 and T(0, 0) = 1.
%F A369117 T(n, k) = [z^k] h(n, n, z) where h(n, v, x) are the modified Lommel polynomials defined by the recurrence h(n, v, x) = 2*(v + n - 1)*z*h(n - 1, v, z) - h(n - 2, v, z) with base values h(-1, v, x) = 0 and h(0, v, x) = 1.
%e A369117 List of coefficients starts:
%e A369117 [0]  1;
%e A369117 [1]  0,   2;
%e A369117 [2] -1,   0,    24;
%e A369117 [3]  0, -16,     0,     480;
%e A369117 [4]  1,   0,  -360,       0,   13440;
%e A369117 [5]  0,  42,     0,  -10752,       0,    483840;
%e A369117 [6] -1,   0,  1728,       0, -403200,         0,   21288960;
%e A369117 [7]  0, -80,     0,   79200,       0, -18247680,          0, 1107025920;
%e A369117 [8]  1,   0, -5280,       0, 4118400,         0, -968647680,       0, 66421555200;
%p A369117 Lommel_h := proc(n) local L, k; if n = 0 then return 1 fi;
%p A369117 h := (n, m, z) -> (GAMMA(n + m)/(GAMMA(n)*(z/2)^m))*hypergeom([(1 - m)/2, -m/2], [n, -m, 1 - n - m], z^2); convert(series(h(n, n, 1/z), z, n + 1), polynom):
%p A369117 seq((-1)^binomial(n-k, 2)*coeff(expand(%), z, k), k = 0..n) end:
%p A369117 for n from 0 to 9 do Lommel_h(n) od;
%p A369117 # Alternative, by recursion:
%p A369117 h := proc(n, v, x) option remember; if n = -1 then 0 elif n = 0 then 1 else
%p A369117 2*(v + n - 1)*z*h(n - 1, v, z) - h(n - 2, v, z) fi end:
%p A369117 for n from 0 to 6 do seq(coeff(h(n, n, z), z, k), k = 0..n) end;
%t A369117 Table[CoefficientList[Expand[ResourceFunction["LommelR"][n, n, 1/z]], z], {n, 0, 9}] // Flatten
%K A369117 sign,tabl
%O A369117 0,3
%A A369117 _Peter Luschny_, Jan 29 2024