A187870
Numerator of the coefficient of x^(2n) in the expansion of 1/x^4 - 1/(3*x^2) - 1/(x^3*arctanh(x)).
Original entry on oeis.org
4, 44, 428, 10196, 10719068, 25865068, 5472607916, 74185965772, 264698472181028, 2290048394728148, 19435959308462817284, 2753151578548809148, 20586893910854623222436, 134344844535611780572028924
Offset: 0
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A187870 := proc(n)
1/x^4 -1/(3*x^2) -1/(x^3*arctanh(x)) ;
coeftayl(%,x=0,2*n) ;
numer(%) ;
end proc:
seq(A187870(n),n=0..10) ; # R. J. Mathar, Sep 21 2011
# Or
seq((-1)^n*numer(coeff(series(1/arctan(x),x,2*n+2),x,2*n+1)),n=1..14); # Peter Luschny, Oct 04 2014
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a[n_] := Sum[(2^(j+1)*Binomial[2*n+3, j]*Sum[(k!*StirlingS1[j+k, j]*StirlingS2[j+1, k])/(j+k)!, {k, 0, j+1}])/(j+1), {j, 0, 2*n+3}]/ (2*n+3); Table[a[n] // Numerator, {n, 0, 13}] (* Jean-François Alcover, Jul 03 2013, after Vladimir Kruchinin's formula in A216272 *)
A216272
Numerators of coefficients in expansion of x/arctan(x)-1 (even powers only).
Original entry on oeis.org
1, -4, 44, -428, 10196, -10719068, 25865068, -5472607916, 74185965772, -264698472181028, 2290048394728148, -19435959308462817284, 2753151578548809148, -20586893910854623222436, 134344844535611780572028924
Offset: 1
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
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# Assuming offset 0:
seq(numer(coeff(series(1/arctan(x),x,2*n+2),x,2*n+1)),n=0..14); # Peter Luschny, Oct 04 2014
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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)
A216272[n_]:=Numerator[b[n]]
nn=20; Numerator[List@@Normal[Series[x/ArcTan[x]-1,{x,0,2nn}]]/.x->1] (* Wolfgang Hintze, Oct 03 2014 *)
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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);
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a(n) = x = y + O(y^(2*n+2)); numerator(polcoeff(x/atan(x)-1, 2*n)) \\ Michel Marcus, Sep 30 2014
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