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%I #11 Apr 30 2013 12:17:02
%S 0,1,1,7,1,43,23,337,22,1091,563,12701,1627,172673,88069,179141,5692,
%T 3141493,1593269,61309987,7759469,8966941,31730711,1474162241,
%U 93044486,7510486741,3788707301,22918154531,2888008157,675037481999,340028535787
%N Numerators of Integral_{x=0..Pi/2} sin(2*n*x)*log(cosec(x)) dx.
%C The integral with sin(2*n*x) is always rational, but the integral with sin((2*n+1)*x) is irrational and is of the form p/q+log(2)/(2*n+1).
%D George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 218.
%e 0, 1/2, 1/2, 7/18, 1/3, 43/150, 23/90, 337/1470, 22/105, 1091/5670, 563/3150, ... = A225122/A225123
%p a:= n-> numer(Re(int(sin(2*n*x)*log(csc(x)), x=0..Pi/2))):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 29 2013
%t a[n_] := Integrate[Sin[2*n*x]*Log[Csc[x]], {x, 0, Pi/2}]; Table[a[n] // Numerator, {n, 0, 30}]
%Y Cf. A225123 (denominators).
%K nonn,easy,frac
%O 0,4
%A _Jean-François Alcover_, Apr 29 2013
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