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%I #73 Feb 21 2022 00:56:28
%S 1,3,45,945,4725,93555,638512875,18243225,162820783125,38979295480125,
%T 1531329465290625,13447856940643125,201919571963756521875,
%U 11094481976030578125,564653660170076273671875,5660878804669082674070015625,31245110285511170603633203125
%N Denominators in Taylor series for cot x.
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.70).
%D G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 19.
%H Seiichi Manyama, <a href="/A036278/b036278.txt">Table of n, a(n) for n = -1..250</a> (terms -1..100 from T. D. Noe)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 75 (4.3.70).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Cotangent.html">Cotangent</a>
%F cot(x) = Sum_{k>=0} (-1)^k B_{2k} 4^k x^(2k-1) / (2k)!.
%F a(n) = denominator(A000182(n)/(4^n-1)), n>0.
%F a(n) = denominator for coth x;
%F coth(x) = W(0)/x -1, W(k) = k+1+2*x-2*x*(k+1)/W(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 27 2011
%F coth(x) = Q(0)/(1+x)/x^2 - 1 where Q(k) = 2*k^3 + (2*x+3)*k^2 + (2*x^2+3*x+1)*k + 2*x^3 + 2*x^2 + x - 2*x^2*(k+1)*(2*k+1)*(x+k)*(x+k+2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Feb 28 2013
%F a(n) = denominator of 4^(n+1)*Zeta(-2*n-1)/(2*n+1)! for n >= 0. - _Peter Luschny_, Jun 20 2013
%F a(n) = denominator(r(n)), with the negative rational numbers r(n) = [x^n]( (cot(sqrt(x))-1/sqrt(x))/sqrt(x)) for n >= 0. - _Wolfdieter Lang_, Oct 07 2016
%e G.f. = 1/x - (1/3)*x - (1/45)*x^3 - (2/945)*x^5 - (1/4725)*x^7 - (2/93555)*x^9 + O(x^11).
%p A036278 := n -> `if`(n<0, 1, denom(4^(n+1)*Zeta(-2*n-1)/(2*n+1)!));
%p seq(A036278(n), n = -1..20); # _Peter Luschny_, Jun 20 2013
%t Denominator[Select[List@@Series[Cot[x],{x,0,20}][[3]],#!=0&]] (* _Harvey P. Dale_, Apr 11 2011 *)
%t a[n_] := (-1)^(n+1)*4^(n+1)*BernoulliB[2*n+2]/(2*n+2)! // Denominator; Table[a[n], {n, -1, 20}] (* _Jean-François Alcover_, Apr 14 2014, after _Peter Luschny_ *)
%o (PARI) apply(r->denominator(r),Vec(1/tan(x))) \\ _Charles R Greathouse IV_, Apr 14 2014
%o (PARI) a(n) = denominator((-1)^(n+1)*4^(n+1)*bernfrac(2*n+2)/(2*n+2)!); \\ _Altug Alkan_, Dec 02 2015
%o (Python)
%o from sympy import bernoulli, factorial
%o def a(n):
%o return ((-4)**(n + 1)*bernoulli(2*n + 2)/factorial(2*n + 2)).denominator()
%o print([a(n) for n in range(-1, 20)]) # _Indranil Ghosh_, Jun 23 2017
%o (Magma) [Denominator((-1)^(n+1)*4^(n+1)*Bernoulli(2*n+2)/Factorial(2*n+2) ): n in [-1..20]]; // _G. C. Greubel_, Jul 03 2019
%o (Sage) [denominator( (-1)^(n+1)*4^(n+1)*bernoulli(2*n+2)/factorial(2*n+2) ) for n in (-1..20)] # _G. C. Greubel_, Jul 03 2019
%o (GAP) List([-1..20], n-> DenominatorRat( (-1)^(n+1)*4^(n+1)* Bernoulli(2*n+2)/Factorial(2*n+2) )) # _G. C. Greubel_, Jul 03 2019
%Y Cf. A000182, A002431 (numerators).
%K nonn,frac,easy,nice
%O -1,2
%A _N. J. A. Sloane_
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