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A216272 Numerators of coefficients in expansion of x/arctan(x)-1 (even powers only).

%I A216272 #45 Apr 12 2017 23:17:49
%S A216272 1,-4,44,-428,10196,-10719068,25865068,-5472607916,74185965772,
%T A216272 -264698472181028,2290048394728148,-19435959308462817284,
%U A216272 2753151578548809148,-20586893910854623222436,134344844535611780572028924
%N A216272 Numerators of coefficients in expansion of x/arctan(x)-1 (even powers only).
%C A216272 Setting the offset to 0 gives the numerators of the odd powers in the expansion of 1/arctan(x). The denominators of the coefficients of the expansion of x/arctan(x) are equal to a shifted sequence A195466. - _Wolfgang Hintze_, Oct 03 2014
%H A216272 G. C. Greubel, <a href="/A216272/b216272.txt">Table of n, a(n) for n = 1..250</a>
%F A216272 a(n) = numerator(((-1)^(n+1)*sum(l=0..2*n-1, (2^(l+1)*(sum(k=0..l+1, (k!*stirling2(l+1,k)*stirling1(l+k,l))/(l+k)!,k,0,l+1))*binomial(2*n-1,l))/(l+1)))/(2*n-1)). - clarified by _Wolfgang Hintze_, Sep 30 2014
%e A216272 Expansion of x/arctan(x)-1: x^2/3 - (4*x^4)/45 + (44*x^6)/945 - (428*x^8)/14175 + (10196*x^10)/467775 - (10719068*x^12)/638512875 + (25865068*x^14)/1915538625 -(5472607916*x^16)/488462349375 + (74185965772*x^18)/7795859096025 - (264698472181028*x^20)/32157918771103125. - _Wolfgang Hintze_, Oct 03 2014
%p A216272 # Assuming offset 0:
%p A216272 seq(numer(coeff(series(1/arctan(x),x,2*n+2),x,2*n+1)),n=0..14); # _Peter Luschny_, Oct 04 2014
%t A216272 b[n_]:=((-1)^(n+1)*Sum[(2^(m+1)*(Sum[(k!*StirlingS2[m+1,k]*StirlingS1[m+k,m])/(m+k)!,{k,0,m+1}]*Binomial [2*n-1,m])/(m+1)),{m,0,2n-1}])/(2*n-1)
%t A216272 A216272[n_]:=Numerator[b[n]]
%t A216272 nn=20; Numerator[List@@Normal[Series[x/ArcTan[x]-1,{x,0,2nn}]]/.x->1] (* _Wolfgang Hintze_, Oct 03 2014 *)
%o A216272 (Maxima) a(n):=((-1)^(n+1)*sum((2^(l+1)*(sum((k!*stirling2(l+1,k)*stirling1(l+k,l))/(l+k)!,k,0,l+1))*binomial(2*n-1,l))/(l+1),l,0,2*n-1))/(2*n-1);
%o A216272 (PARI) a(n) = x = y + O(y^(2*n+2)); numerator(polcoeff(x/atan(x)-1, 2*n)) \\ _Michel Marcus_, Sep 30 2014
%Y A216272 Cf. A195466.
%K A216272 sign,frac
%O A216272 1,2
%A A216272 _Vladimir Kruchinin_, Mar 16 2013
%E A216272 Definition corrected by _Wolfgang Hintze_, Sep 30 2014