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A369585 Table read by rows. T(n, k) = [z^k] h(n, 1, z) where h(n, v, z) are the modified Lommel polynomials (A369117).

%I A369585 #13 Mar 14 2025 09:34:27
%S A369585 1,0,2,-1,0,8,0,-8,0,48,1,0,-72,0,384,0,18,0,-768,0,3840,-1,0,288,0,
%T A369585 -9600,0,46080,0,-32,0,4800,0,-138240,0,645120,1,0,-800,0,86400,0,
%U A369585 -2257920,0,10321920,0,50,0,-19200,0,1693440,0,-41287680,0,185794560
%N A369585 Table read by rows. T(n, k) = [z^k] h(n, 1, z) where h(n, v, z) are the modified Lommel polynomials (A369117).
%H A369585 David Dickinson, <a href="https://doi.org/10.1090/S0002-9939-1954-0086897-8">On Lommel and Bessel polynomials</a>, Proc. AMS 5 (1954) 946-956.
%H A369585 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LommelPolynomial.html">Lommel Polynomial</a>.
%F A369585 T(n, k) = [z^k] 2*n*z*p(n-1, z) - p(n-2, z) where p(-1, z) = 0 and p(0, z) = 1.
%F A369585 T(n, k) = (-1)^k * [z^k] h(n, -n, z) where h(n, v, z) are the modified Lommel polynomials (A369117).
%e A369585 The list of coefficients starts:
%e A369585   [0]  1
%e A369585   [1]  0,   2
%e A369585   [2] -1,   0,    8
%e A369585   [3]  0,  -8,    0,   48
%e A369585   [4]  1,   0,  -72,    0,   384
%e A369585   [5]  0,  18,    0, -768,     0,    3840
%e A369585   [6] -1,   0,  288,    0, -9600,       0,    46080
%e A369585   [7]  0, -32,    0, 4800,     0, -138240,        0, 645120
%e A369585   [8]  1,   0, -800,    0, 86400,       0, -2257920,      0, 10321920
%p A369585 p := proc(n,  x) option remember; if n = -1 then 0 elif n = 0 then 1 else
%p A369585 2*n*z*p(n - 1, z) - p(n - 2, z) fi end:
%p A369585 seq(seq(coeff(p(n, z), z, k), k = 0..n), n = 0..9);
%t A369585 Table[CoefficientList[Expand[ResourceFunction["LommelR"][n, 1, 1/z]], z], {n, 0, 8}] // MatrixForm
%Y A369585 Diagonals include: A000165 (main diagonal), A014479, A286725.
%Y A369585 Columns include bisections of: A001105, A254371.
%Y A369585 Cf. A093985 (row sums), A036243 (abs row sums), A369117.
%K A369585 sign,tabl
%O A369585 0,3
%A A369585 _Peter Luschny_, Jan 30 2024