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A216272
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Numerators of coefficients in expansion of x/arctan(x)-1 (even powers only).
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3
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1, -4, 44, -428, 10196, -10719068, 25865068, -5472607916, 74185965772, -264698472181028, 2290048394728148, -19435959308462817284, 2753151578548809148, -20586893910854623222436, 134344844535611780572028924
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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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MAPLE
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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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MATHEMATICA
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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)
nn=20; Numerator[List@@Normal[Series[x/ArcTan[x]-1, {x, 0, 2nn}]]/.x->1] (* Wolfgang Hintze, Oct 03 2014 *)
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PROG
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(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);
(PARI) 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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CROSSREFS
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KEYWORD
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sign,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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