A217766 - OEIS

%I #35 Mar 13 2019 12:27:58

%S 0,2,31,1209,87510,10062642,1676297196,380613039300,112785012934704,

%T 42220061283665808,19466179705605460320,10832183496342326864160,

%U 7154687325911822697398400,5531732531984974533825018240,4947671342477051367102277159680,5067624845854754327998998304876800

%N Numerators for a rational approximation to Euler constant.

%C a(n)/A217767(n) converges to Euler's constant.

%C 0 < A217766(n)/A217767(n)-gamma < 2*Pi*exp(-2*sqrt(2n))(1+O(n^(-1/2))).

%D A. I. Aptekarev (Editor), Rational approximants for Euler's constant and recurrence relations, Collected papers, Sovrem. Probl. Mat. ("Current Problems in Mathematics") Vol. 9, MIAN (Steklov Institute), Moscow (2007), 84pp (Russian)

%H G. C. Greubel, <a href="/A217766/b217766.txt">Table of n, a(n) for n = 0..223</a>

%H Kh. Hessami Pilehrood, T. Hessami Pilehrood, <a href="https://doi.org/10.1016/j.jnt.2012.08.016">On a continued fraction expansion for Euler's constant</a>, Journal of Number Theory, 133 (2013) 769--786.

%H D. N. Tulyakov, <a href="https://doi.org/10.4213/mzm5260">A system of recurrence relations for rational approximations of the Euler constant</a>, (Russian) Mat. Zametki 85 (2009), No. 5 , 782-787. <a href="https://doi.org/10.1134/S0001434609050150">Translation</a>: Mathematical Notes 85 (2009), No. 5, 746-750.

%F a(n) = Sum_{k=0..n} binomial(n,k)^2 *(n+k)!*(H(n+k)+2*H(n-k)-2*H(k)) where H(n) is the n-th harmonic number. (Pilehrood)

%F (16*n - 15)*a(n+1) = (128*n^3 + 40*n^2 - 82*n - 45)*a(n) - n^2*(256*n^3 -240*n^2 +64*n-7)*a(n-1) +(16*n + 1)*n^2*(n - 1)^2*a(n-2), (the integrality has been proved by Tulyakov).

%e G.f. = 2*x + 31*x^2 + 1209*x^3 + 87510*x^4 + 10062642*x^5 + ...

%t Table[ Sum[ Binomial[n, k]^2 (n + k)! (HarmonicNumber[n + k] + 2 HarmonicNumber[n - k] - 2 HarmonicNumber[k]), {k, 0, n}], {n, 0, 20}]

%o (PARI) {a(n) = my(H = k->sum(i=1, k, 1/i)); sum(k=0, n, binomial(n, k)^2 * (n+k)! * (H(n+k) + 2*H(n-k) - 2*H(k)))}; /* _Michael Somos_, Nov 25 2016 */

%Y Cf. A217767 (denominators).

%K nonn,frac

%O 0,2

%A _Juan Arias-de-Reyna_, Mar 24 2013