|
| |
|
|
A363515
|
|
Numerator of log(2) + (-1/4)^n*Integral_{x=0..1} (1 - x)^(4*n+2)/(1 + x^2) dx.
|
|
2
|
|
|
|
1, 79, 14087, 3990557, 217474889, 10326377909, 19001942777579, 3306285643032971, 3279846716611480357, 121354235196693865579, 19902098013482397470501, 1711580361934007500382731, 9009759106282339175994464009, 59689653955233943488755746919, 3820137854975012405338172218301
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
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
|
|
|
|
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...
...
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
|
|
|
|
STATUS
|
approved
|
| |
|
|
