OFFSET
0,2
COMMENTS
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
KEYWORD
nonn,frac
AUTHOR
Juan Arias-de-Reyna, Mar 24 2013
STATUS
approved