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

1, 0, 2, -1, 0, 24, 0, -16, 0, 480, 1, 0, -360, 0, 13440, 0, 42, 0, -10752, 0, 483840, -1, 0, 1728, 0, -403200, 0, 21288960, 0, -80, 0, 79200, 0, -18247680, 0, 1107025920, 1, 0, -5280, 0, 4118400, 0, -968647680, 0, 66421555200, 0, 130, 0, -349440, 0, 242161920, 0, -59041382400, 0, 4516665753600

Offset

0,3

Comments

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.

References

Eugen von Lommel, Zur Theorie der Bessel'schen Functionen, Math. Ann. 4, 103-116 (1871).

Links

Table of n, a(n) for n=0..54.

David Dickinson, On Lommel and Bessel polynomials, AMS Proceedings 1954.

Eric Weisstein's World of Mathematics, Lommel Polynomial.

Formula

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.

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.

Example

List of coefficients starts:
[0]  1;
[1]  0,   2;
[2] -1,   0,    24;
[3]  0, -16,     0,     480;
[4]  1,   0,  -360,       0,   13440;
[5]  0,  42,     0,  -10752,       0,    483840;
[6] -1,   0,  1728,       0, -403200,         0,   21288960;
[7]  0, -80,     0,   79200,       0, -18247680,          0, 1107025920;
[8]  1,   0, -5280,       0, 4118400,         0, -968647680,       0, 66421555200;

Maple

Lommel_h := proc(n) local L, k; if n = 0 then return 1 fi;
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):
seq((-1)^binomial(n-k, 2)*coeff(expand(%), z, k), k = 0..n) end:
for n from 0 to 9 do Lommel_h(n) od;
# Alternative, by recursion:
h := proc(n, v, x) option remember; if n = -1 then 0 elif n = 0 then 1 else
2*(v + n - 1)*z*h(n - 1, v, z) - h(n - 2, v, z) fi end:
for n from 0 to 6 do seq(coeff(h(n, n, z), z, k), k = 0..n) end;

Mathematica

Table[CoefficientList[Expand[ResourceFunction["LommelR"][n, n, 1/z]], z], {n, 0, 9}] // Flatten

Crossrefs

Sequence in context: A269946 A009829 A202697 * A268436 A249570 A051652

Adjacent sequences: A369114 A369115 A369116 * A369118 A369119 A369120

Keyword

sign,tabl

Author

Peter Luschny, Jan 29 2024

Status

approved