A195466 Denominator of the coefficient of x^(2n) in expansion of 1/x^4 - 1/(3*x^2) - 1/(x^3*arctanh(x)).

45, 945, 14175, 467775, 638512875, 1915538625, 488462349375, 7795859096025, 32157918771103125, 316985199315159375, 3028793579456347828125, 478230565177318078125, 3952575621190533915703125, 28304394023345413370350078125, 7217620475953080409439269921875, 21652861427859241228317809765625

Offset: 0

Author

R. J. Mathar, Sep 21 2011

Keywords

• nonn
• frac

Comments

Prepending 3 to the data gives the denominators of the odd powers in the expansion of 1/arctan(x). - Peter Luschny, Oct 04 2014

Crossrefs

Cf. A187870 (numerator).

Programs

Maple

A195466 := proc(n)
        1/x^4 -1/(3*x^2) -1/(x^3*arctanh(x)) ;
        coeftayl(%,x=0,2*n) ;
        denom(%) ;
end proc
seq(A195466(n),n=0..15) ;
# Or
seq(denom(coeff(series(1/arctan(x),x,2*n+2),x,2*n+1)),n=1..16); # Peter Luschny, Oct 04 2014

Mathematica

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] // Denominator, {n, 0, 15}] (* Jean-François Alcover, Jul 03 2013, after Vladimir Kruchinin's formula in A216272 *)

Formula

1/x^4 - 1/(3x^2) - 1/(x^3*arctanh x) = 4/45 + 44*x^2/945 + 428*x^4/14175 + 10196*x^6/467775 + ...