OFFSET
0,2
COMMENTS
From M. F. Hasler, Jul 07 2023: (Start)
Equivalently, numerator of Sum c(n,k)/(k+1), where Sum c(n,k)*x^k = ((1 - x)^(4*n+2)/(-4)^n + 2*x)/(1 + x^2), a polynomial: The integrand (with factor (-1/4)^n) plus 2*x/(1 + x^2) is a polynomial that is easily integrated to yield the fraction a(n)/A363516(n), while Integral(-2*x/(1 + x^2)) = -log(1 + x^2) cancels the log(2).
Since the integrand/integral as a whole is less than 1/4^n in absolute value, it tends to zero and the fraction tends to log(2). (End)
LINKS
Mathematica Stack Exchange, How to prove using Mathematica that the sequence converges to log(2)?
FORMULA
Numerator of log(2) + HypergeometricPFQ([1/2, 1, 1], [2*(1 + n), 5/2 + 2*n], -1)/((3 + 4*n)*(-4)^n).
Limit_{n->oo} a(n)/A363516(n) = log(2).
EXAMPLE
n a(n)/A363516(n) approximate value
- ------------------- ------------------
0 1 1
1 79/120 0.6583333333...
2 14087/20160 0.6987599206...
3 3990557/5765760 0.6921129218...
4 217474889/313657344 0.6933518158...
...
From M. F. Hasler, Jul 07 2023: (Start)
Let f[n] = (-1/4)^n*(1 - x)^(4*n+2)/(1 + x^2), the rational fraction to be integrated from 0 to 1. We have:
f[0] = 1 - 2*x/(1 + x^2), with primitive F[0] = x/2 - log(1 + x^2), whence an integral equal to 1/2 - log(2).
f[1] = -x^4/4 + (3/2)*x^3 - (7/2)*x^2 + (7/2)*x - 1/4 - 2*x/(1 + x^2), and the term-wise integration of the polynomial part gives -1/20 + 3/8 - 7/6 + 7/4 - 1/4 = 79/120, whence a(1) = 79 and A363516(1) = 120.
f[2] = (1/16)*x^8 - (5/8)*x^7 + (11/4)*x^6 - (55/8)*x^5 + (83/8)*x^4 - (71/8)*x^3 + (11/4)*x^2 + (11/8)*x + 1/16 - 2*x/(1 + x^2), so the integration gives 1/144 - 5/64 + 11/28 - 55/48 + 83/40 - 71/32 + 11/12 + 11/16 + 1/16 - log(2) = 14087/20160 - log(2), whence a(2) = 14087 and A363516(2) = 20160, etc. (End)
MATHEMATICA
Numerator[Simplify[Table[Log[2]+(-1)^n Integrate[(1-x)^(4n+2)/(1+x^2), {x, 0, 1}]/4^n, {n, 0, 14}]]]
PROG
(PARI) A363515(n) = numerator(subst(intformal(((1-x)^(4*n+2)/(-4)^n+2*x)/(1+x^2)), x, 1)) \\ The argument of intformal is a polynomial that is trivially integrated over [0, 1]. - M. F. Hasler, Jul 07 2023
CROSSREFS
KEYWORD
nonn,frac,less
AUTHOR
Alexander R. Povolotsky and Stefano Spezia, Jun 07 2023
STATUS
approved