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%I #45 Dec 26 2019 19:00:24
%S 1,-2,26,-502,7102,-44834,295272982,-122850554,19437784634,
%T -83457787614326,13505836484182762,-83261125331410322,
%U 1230729837542663167546,-279990740971966317602,31893076454808467404426
%N Numerators of coefficients in expansion of x/((x^2+1)*arctan(x)), even powers only.
%C The denominators are given in A225149.
%H Vincenzo Librandi, <a href="/A216254/b216254.txt">Table of n, a(n) for n = 0..100</a>
%H D. Kruchinin and V. Kruchinin, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL18/Kruchinin/kruch9.html">A Generating Function for the Diagonal T2n,n in Triangles</a>, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.
%F a(n) = numerator((-1)^n*sum(l=0..2*n, 2^l * (sum(k=0..l, (k!*stirling2(l,k) * stirling1(l+k,l)) / (l+k)!)) * binomial(2*n,l))).
%F a(n) = numerator(b(n)), where b(n) = (-1)^n*(1-1/(2*n+1)-sum(i=1..n-1, b(i)*(-1)^i/(2*(n-i)+1))), b(0)=1. [_Vladimir Kruchinin_, Aug 29 2013]
%e x/((x^2+1)*atan(x)) = 1 - 2/3*x^2 + 26/45*x^4 - 502/945*x^6 + 7102/14175*x^8 - 44834/93555*x^10 + 295272982/638512875*x^12 - 122850554/273648375*x^14 + ...
%t a[n_] := (-1)^n*Sum[2^l*(Sum[(k!*StirlingS2[l, k]*StirlingS1[l+k, l])/(l+k)!, {k, 0, l}])* Binomial[2*n, l], {l, 0, 2*n}]; Table[a[n] // Numerator, {n, 0, 14}] (* _Jean-François Alcover_, Apr 30 2013, translated from Maxima *)
%t Take[CoefficientList[Series[x/((x^2+1)ArcTan[x]),{x,0,30}],x],{1,-1,2}]//Numerator (* _Harvey P. Dale_, Dec 26 2019 *)
%o (Maxima) a(n):=(-1)^n*sum(2^l*(sum((k!*stirling2(l,k) * stirling1(l+k,l))/(l+k)!,k,0,l)) * binomial(2*n,l),l,0,2*n).
%o (PARI) x='x+O('x^66); v=Vec(x/((x^2+1)*atan(x))); vector(#v\2,n,numerator(v[2*n-1])) \\ _Joerg Arndt_, Apr 29 2013
%K sign,frac
%O 0,2
%A _Vladimir Kruchinin_, Mar 15 2013
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