A036278 Denominators in Taylor series for cot x.

1, 3, 45, 945, 4725, 93555, 638512875, 18243225, 162820783125, 38979295480125, 1531329465290625, 13447856940643125, 201919571963756521875, 11094481976030578125, 564653660170076273671875, 5660878804669082674070015625, 31245110285511170603633203125

Offset

-1,2

References

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).

G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.

H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 19.

Links

Seiichi Manyama, Table of n, a(n) for n = -1..250 (terms -1..100 from T. D. Noe)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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).

Eric Weisstein's World of Mathematics, Cotangent

Formula

cot(x) = Sum_{k>=0} (-1)^k B_{2k} 4^k x^(2k-1) / (2k)!.

a(n) = denominator(A000182(n)/(4^n-1)), n>0.

a(n) = denominator for coth x;

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

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

a(n) = denominator of 4^(n+1)*Zeta(-2*n-1)/(2*n+1)! for n >= 0. - Peter Luschny, Jun 20 2013

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

Example

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).

Maple

A036278 := n -> `if`(n<0, 1, denom(4^(n+1)*Zeta(-2*n-1)/(2*n+1)!));
seq(A036278(n), n = -1..20);  # Peter Luschny, Jun 20 2013

Mathematica

Denominator[Select[List@@Series[Cot[x], {x, 0, 20}][[3]], #!=0&]]  (* Harvey P. Dale, Apr 11 2011 *)
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 *)

Prog

(PARI) apply(r->denominator(r), Vec(1/tan(x))) \\ Charles R Greathouse IV, Apr 14 2014
(PARI) a(n) = denominator((-1)^(n+1)*4^(n+1)*bernfrac(2*n+2)/(2*n+2)!); \\  Altug Alkan, Dec 02 2015
(Python)
from sympy import bernoulli, factorial
def a(n):
    return ((-4)**(n + 1)*bernoulli(2*n + 2)/factorial(2*n + 2)).denominator()
print([a(n) for n in range(-1, 20)]) # Indranil Ghosh, Jun 23 2017
(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
(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
(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

Crossrefs

Cf. A000182, A002431 (numerators).

Sequence in context: A298799 A202437 A008931 * A225149 A154289 A171080

Adjacent sequences: A036275 A036276 A036277 * A036279 A036280 A036281

Keyword

nonn,frac,easy,nice

Author

N. J. A. Sloane

Status

approved