OFFSET
0,2
COMMENTS
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.
LINKS
Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
A. Kreinin, Combinatorial properties of the Mills Ratio, arXiv:1405.5852 [math.CO], 2014.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
FORMULA
a(n) = ((2*n+6)!! - 3*(2*n+5)!! + (2*n+3)!!)/6, n>=0.
MATHEMATICA
Table[((2 n + 6)!! - 3 (2 n + 5)!! + (2 n + 3)!!)/6, {n, 0, 12}] (* Michael De Vlieger, Oct 27 2015 *)
PROG
(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;
vector(20, n, a(n-1)) \\ Altug Alkan, Oct 16 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
Alexander Kreinin, Oct 16 2015
STATUS
approved