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%I #33 Sep 08 2022 08:46:04
%S 1,3,50,2022,148824,17254920,2886892560,657047386800,194964822138240,
%T 73042276012030080,33693790560582700800,18755069649902783366400,
%U 12390207483469555200384000,9580861371340114269711897600,8570002001492431798612092979200
%N Denominators for a rational approximation to Euler constant.
%C A217766(n)/a(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="/A217767/b217767.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.
%F a(n) = Sum_{k=0}^n binomial(n,k)^2 (n+k)! (Pilehrood)
%F (16n - 15)*a(n+1) = (128n^3 + 40n^2 - 82n - 45)*a(n) - n^2(256n^3 -240n^2 +64n-7)*a(n-1) +(16n + 1)n^2(n - 1)^2*a(n-2), with a(0)=1; a(1)=3; a(2)=50.
%F a(n) ~ (2*n)^(2*n+1/4)*exp(sqrt(2*n)-2*n-3/8) * (1 + 97/(96*sqrt(2*n))). - _Vaclav Kotesovec_, Jul 26 2013
%F 0 = +a(n) * (+a(n+1) * (+2*a(n+2) + 12*a(n+3) - a(n+4)) + a(n+2) * (+6*a(n+2) + 44*a(n+3) - 5*a(n+4)) + 9*a(n+3)^2) + a(n+1) * (+a(n+1) * (+6*a(n+2) + 36*a(n+3) - 3*a(n+4)) + a(n+2) * (+17*a(n+2) + 130*a(n+3) - 16*a(n+4)) + 32*a(n+3)^2) + a(n+2)^2 * (-3*a(n+2) - 16*a(n+3)). - _Michael Somos_, Mar 25 2014
%F 0 = +a(n) * (-47784*a(n+3) + 33108*a(n+4) + 1150942*a(n+5) - 36961*a(n+6) + 183*a(n+7)) + a(n+1) * (-262812*a(n+3) + 339402*a(n+4) + 7523706*a(n+5) - 347559*a(n+6) + 2368*a(n+7)) + a(n+2) * (-302632*a(n+3) + 1024881*a(n+4) + 13312395*a(n+5) - 1017760*a(n+6) + 10160*a(n+7))+ a(n+3) * (+51766*a(n+3) + 804453*a(n+4) + 4760128*a(n+5) - 898320*a(n+6) + 14464*a(n+7)) + a(n+4) * (-141280*a(n+4) - 500384*a(n+5) - 57856*a(n+6)) + a(n+5)*(+43392*a(n+5)). - _Michael Somos_, Mar 25 2014
%t Table[Sum[Binomial[n, k]^2 (n + k)!, {k, 0, n}], {n, 1, 20}]
%o (PARI) for(n=0,20, print1(sum(k=0,n, binomial(n,k)^2*(n+k)!), ", ")) \\ _G. C. Greubel_, Aug 10 2018
%o (Magma) [(&+[Binomial(n,k)^2*Factorial(n+k): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 10 2018
%Y Cf. A217766 (numerators).
%K nonn,frac
%O 0,2
%A _Juan Arias-de-Reyna_, Mar 24 2013
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