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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 September 19 20:02 EDT 2020. Contains 337182 sequences. (Running on oeis4.)