%I #31 Aug 24 2018 21:09:05
%S 1,6,9,2,9,9,2,4,6,8,4,1,3,6,0,1,2,4,4,6,7,8,0,1,3,8,3,4,8,9,8,1,0,8,
%T 7,0,8,0,7,8,6,9,8,6,7,1,5,6,8,0,7,2,3,4,9,5,6,8,8,0,1,5,7,7,8,9,4,7,
%U 6,4,3,7,2,1,3,1,9,8,7,9,8,7,2,7,9,1,8,7,3,6,3,9,6,3,5,4,4,9,4,2
%N Decimal expansion of Integral_{x=0..Pi/2} x^3*cosec(x) dx.
%C The simpler Integral_{x=0..Pi/2} x*cosec(x) dx evaluates as 2*Catalan.
%H G. C. Greubel, <a href="/A225125/b225125.txt">Table of n, a(n) for n = 1..10000</a>
%H StackExchange, <a href="http://math.stackexchange.com/questions/302087/a-integral-with-polygamma">An integral with PolyGamma.</a>
%F Equals 3*Catalan*Pi^2/2-1/128*(polygamma(3, 1/4)-polygamma(3, 3/4)).
%e 1.6929924684136012446780138348981087080786986715680723495688...
%t 3*Catalan*Pi^2/2-1/128*(PolyGamma[3, 1/4]-PolyGamma[3, 3/4]); (* or *)
%t 3*Catalan*Pi^2/2-3/64*(Zeta[4, 1/4]-Zeta[4, 3/4]) // RealDigits[#, 10, 100] & // First
%o (PARI) 3*Catalan*Pi^2/2-3/64*(zetahurwitz(4,1/4)-zetahurwitz(4,3/4)) \\ _Charles R Greathouse IV_, Jan 31 2018
%Y Cf. A006752, A221209.
%K nonn,cons,easy
%O 1,2
%A _Jean-François Alcover_, Apr 30 2013