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 A217767 Denominators for a rational approximation to Euler constant. 2
 1, 3, 50, 2022, 148824, 17254920, 2886892560, 657047386800, 194964822138240, 73042276012030080, 33693790560582700800, 18755069649902783366400, 12390207483469555200384000, 9580861371340114269711897600, 8570002001492431798612092979200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A217766(n)/a(n) converges to Euler's constant. 0 < A217766(n)/A217767(n)-gamma < 2*Pi*exp(-2*sqrt(2n))(1+O(n^(-1/2))). REFERENCES 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). LINKS G. C. Greubel, Table of n, a(n) for n = 0..223 Kh. Hessami Pilehrood, T. Hessami Pilehrood, On a continued fraction expansion for Euler's constant, Journal of Number Theory, 133 (2013) 769--786. FORMULA a(n) = Sum_{k=0}^n binomial(n,k)^2 (n+k)! (Pilehrood) (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. 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 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 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 MATHEMATICA Table[Sum[Binomial[n, k]^2 (n + k)!, {k, 0, n}], {n, 1, 20}] PROG (PARI) for(n=0, 20, print1(sum(k=0, n, binomial(n, k)^2*(n+k)!), ", ")) \\ G. C. Greubel, Aug 10 2018 (Magma) [(&+[Binomial(n, k)^2*Factorial(n+k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 10 2018 CROSSREFS Cf. A217766 (numerators). Sequence in context: A245141 A203239 A279970 * A185157 A078674 A071094 Adjacent sequences: A217764 A217765 A217766 * A217768 A217769 A217770 KEYWORD nonn,frac AUTHOR Juan Arias-de-Reyna, Mar 24 2013 STATUS approved

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Last modified June 3 14:22 EDT 2023. Contains 363116 sequences. (Running on oeis4.)