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A263384
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Fourth column of the matrix of polynomial coefficients of the rational approximation to Mill's ratio.
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0
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1, 14, 185, 2640, 41685, 729330, 14073885, 297693900, 6859400625, 171172905750, 4601737965825, 132643472761800, 4082080279402125, 133614981594344250, 4635763624512145125, 169957871025837394500
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = ((2*n+6)!! - 3*(2*n+5)!! + (2*n+3)!!)/6, n>=0.
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MATHEMATICA
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Table[((2 n + 6)!! - 3 (2 n + 5)!! + (2 n + 3)!!)/6, {n, 0, 12}] (* Michael De Vlieger, Oct 27 2015 *)
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PROG
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(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;
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CROSSREFS
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Columns of the matrix [q_{k,m}] include: A000165 (m=1), A129890 (m=2), A035101 (m=3), this sequence (m=4).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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