cp's OEIS Frontend

A263384 Fourth column of the matrix of polynomial coefficients of the rational approximation to Mill's ratio.

%I A263384 #43 Dec 02 2017 02:25:30
%S A263384 1,14,185,2640,41685,729330,14073885,297693900,6859400625,
%T A263384 171172905750,4601737965825,132643472761800,4082080279402125,
%U A263384 133614981594344250,4635763624512145125,169957871025837394500
%N A263384 Fourth column of the matrix of polynomial coefficients of the rational approximation to Mill's ratio.
%C A263384 Rational approximations, Q_{k-1}(t)/P_k(t), to Mill's ratio, R(t)=(1-Phi(t))/f(t), where Phi(t) is the standard normal distribution function and f(t) is the standard normal density, were discovered by Laplace, who computed the first four polynomials. Thirty years later, Jacobi derived recurrence relations for these polynomials and analyzed some of their analytical properties. The coefficients q_{k,m} of Q_k(t) form a matrix, of which this is the fourth column. The double generating function for the polynomials Q_k(t) is computed in A. Kreinin (see Links). The coefficients q_{k,m} are described by the triangular array A180048.
%H A263384 Selden Crary, Richard Diehl Martinez, Michael Saunders, <a href="https://arxiv.org/abs/1707.00705">The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters</a>, arXiv:1707.00705 [stat.ME], 2017, Table 2.
%H A263384 A. Kreinin, <a href="http://arxiv.org/abs/1405.5852">Combinatorial properties of the Mills Ratio</a>, arXiv:1405.5852 [math.CO], 2014.
%H A263384 Alexander Kreinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Kreinin/kreinin4.html">Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity</a>, Journal of Integer Sequences, 19 (2016), #16.6.2.
%F A263384 a(n) = ((2*n+6)!! - 3*(2*n+5)!! + (2*n+3)!!)/6, n>=0.
%t A263384 Table[((2 n + 6)!! - 3 (2 n + 5)!! + (2 n + 3)!!)/6, {n, 0, 12}] (* _Michael De Vlieger_, Oct 27 2015 *)
%o A263384 (PARI) a(n)=(prod(k=1, n+3, 2*k)-3*prod(k=1, n+3,(2*k-1))+prod(k=1, n+2, 2*k-1))/6;
%o A263384 vector(20, n, a(n-1)) \\ _Altug Alkan_, Oct 16 2015
%Y A263384 Columns of the matrix [q_{k,m}] include: A000165 (m=1), A129890 (m=2), A035101 (m=3), this sequence (m=4).
%Y A263384 Cf. A180048.
%K A263384 nonn,easy,nice
%O A263384 0,2
%A A263384 _Alexander Kreinin_, Oct 16 2015