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

1, -4, 44, -428, 10196, -10719068, 25865068, -5472607916, 74185965772, -264698472181028, 2290048394728148, -19435959308462817284, 2753151578548809148, -20586893910854623222436, 134344844535611780572028924

Offset: 1

Author

Vladimir Kruchinin, Mar 16 2013

Keywords

• sign
• frac

Comments

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

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..250

Crossrefs

Cf. A195466.

Programs

Maple

# 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

Mathematica

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 *)

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

Formula

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

Extensions

Definition corrected by Wolfgang Hintze, Sep 30 2014