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A193717 Decimal expansion of Pi^4*log(2)/64 - 9*Pi^2*zeta(3)/64 + 93*zeta(5)/128. 3
1, 4, 0, 0, 2, 4, 1, 0, 1, 7, 0, 6, 8, 5, 2, 3, 1, 7, 1, 0, 0, 2, 7, 0, 5, 7, 8, 8, 7, 5, 5, 3, 5, 0, 7, 5, 3, 2, 2, 4, 2, 8, 2, 1, 2, 7, 8, 5, 7, 7, 0, 5, 0, 8, 9, 8, 8, 1, 8, 5, 9, 6, 3, 1, 4, 1, 1, 6, 2, 7, 7, 1, 4, 6, 3, 7, 0, 5, 9, 7, 0, 2, 3, 0, 4, 9, 0, 7, 6, 1, 1, 0, 2, 6, 6, 3, 0, 9, 0, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The absolute value of the integral {x=0..Pi/2} x^3*log(sin(x )) dx or (d^3/da^3 (integral {x=0..Pi/2} sin(ax )*log(sin(x )) dx)) at a=0. The absolute value of (sum {n=1..infinity} (limit { a -> 0} (d^3/da^3 ((1-cos((a+2n)*Pi/2))/n/(a+2n)))))-(Pi/2)^4*log(2)/4. [Seiichi Kirikami and Peter J. C. Moses]
REFERENCES
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, series and Products, 4th edition, 1.441.2, log(sin(x))=-(sum {1..infinity} cos(2nx)/n)-log(2).
LINKS
S. Koyama and N. Kurokawa, Euler’s integrals and multiple sine functions, Proc. Amer. Math. Soc. 133(2005), 1257-1265.
FORMULA
Equals A092425*A002162/64-9*A002388*A002117/64+93*A013663/128.
EXAMPLE
-0.14002410170685231710...
MATHEMATICA
RealDigits[N[(2 Pi^4 Log[2] - 18 Pi^2 Zeta[3] + 93 Zeta[5]) / 128, 105]][[1]]
PROG
(PARI) Pi^4*log(2)/64 - 9*Pi^2*zeta(3)/64 + 93*zeta(5)/128 \\ Michel Marcus, Oct 25 2017
CROSSREFS
Sequence in context: A292900 A177893 A058305 * A020808 A198364 A325492
KEYWORD
cons,nonn
AUTHOR
Seiichi Kirikami, Aug 03 2011
STATUS
approved

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Last modified February 23 01:37 EST 2024. Contains 370265 sequences. (Running on oeis4.)