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A363516
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Denominator of log(2) + (-1/4)^n*Integral_{x=0..1} (1 - x)^(4*n+2)/(1 + x^2) dx.
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1
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1, 120, 20160, 5765760, 313657344, 14898723840, 27413651865600, 4769975424614400, 4731815621217484800, 175077177985046937600, 28712657189547697766400, 2469288518301102007910400, 12998334760337000969640345600, 86113967787232631423867289600, 5511293938382888411127506534400
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OFFSET
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0,2
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COMMENTS
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Equivalently, denominator of Sum c(n,k)/(k+1), where ((1 - x)^(4*n+2)/(-4)^n + 2*x)/(1 + x^2) = Sum c(n,k)*x^k, a polynomial. In other words, the integrand (with factor (-1/4)^n) plus 2*x/(1 + x^2) is a polynomial that can easily be term-wise integrated to yield the fraction A363515(n)/a(n), while antiderivative(-2*x/(1 + x^2)) = -log(1 + x^2) cancels the log(2). - M. F. Hasler, Jul 07 2023
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LINKS
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FORMULA
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Denominator of log(2) + HypergeometricPFQ([1/2, 1, 1], [2*(1 + n), 5/2 + 2*n], -1)/((3 + 4*n)*(-4)^n).
Limit_{n->oo} A363515(n)/a(n) = log(2).
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EXAMPLE
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n A363515(n)/a(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), and a(0) = 2 (denominator).
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 A363515(1) = 79 and a(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 A363515(2) = 14087 and a(2) = 20160, etc. (End)
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MATHEMATICA
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Denominator[Simplify[Table[Log[2]+(-1)^n Integrate[(1-x)^(4n+2)/(1+x^2), {x, 0, 1}]/4^n, {n, 0, 14}]]]
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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