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

1, 14, 185, 2640, 41685, 729330, 14073885, 297693900, 6859400625, 171172905750, 4601737965825, 132643472761800, 4082080279402125, 133614981594344250, 4635763624512145125, 169957871025837394500

Offset: 0

Author

Alexander Kreinin, Oct 16 2015

Keywords

• nonn
• easy
• nice

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.

Crossrefs

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

Cf. A180048.

Programs

Mathematica

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

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

Formula

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