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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
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
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
KEYWORD
nonn,frac
AUTHOR
Juan Arias-de-Reyna, Mar 24 2013
STATUS
approved