A263384 - OEIS

%I #43 Dec 02 2017 02:25:30

%S 1,14,185,2640,41685,729330,14073885,297693900,6859400625,

%T 171172905750,4601737965825,132643472761800,4082080279402125,

%U 133614981594344250,4635763624512145125,169957871025837394500

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

%C 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 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 A. Kreinin, <a href="http://arxiv.org/abs/1405.5852">Combinatorial properties of the Mills Ratio</a>, arXiv:1405.5852 [math.CO], 2014.

%H 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 a(n) = ((2*n+6)!! - 3*(2*n+5)!! + (2*n+3)!!)/6, n>=0.

%t Table[((2 n + 6)!! - 3 (2 n + 5)!! + (2 n + 3)!!)/6, {n, 0, 12}] (* _Michael De Vlieger_, Oct 27 2015 *)

%o (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 vector(20, n, a(n-1)) \\ _Altug Alkan_, Oct 16 2015

%Y Columns of the matrix [q_{k,m}] include: A000165 (m=1), A129890 (m=2), A035101 (m=3), this sequence (m=4).

%Y Cf. A180048.

%K nonn,easy,nice

%O 0,2

%A _Alexander Kreinin_, Oct 16 2015