OFFSET
-1,4
COMMENTS
Can be written as numerators of multiples of Bernoulli numbers.
From Wolfdieter Lang, Jun 12 2017: (Start)
cot(x) = Sum_{k>=0} r(k-1)*x^(2*k-1), with the rationals r(n) = a(n)/A036278(n), for n >= -1, for 0 < |x| < Pi.
coth(x) = Sum_{k>=0} (-1)^k*r(k-1)*x^(2*k-1), for 0 < |x| < Pi.
Exercise 2., ch. VI, in Whittaker-Watson, p. 122: 4*Integral_{y=0..infinity} sin(x*y)/(exp(2*Pi*y)-1) dy = coth(x/2) - 2/x. Attributed to Legendre. (End)
Let c(1) = 1/3, c(n) = (Sum_{k=1..n-1} c(k)*c(n-k))/(2*n+1) = -(-1)^n * 2^(2*n) * bernoulli(2*n) / (2*n)!. Then f(x) := 1 - x * cot(x) = Sum_{n>=1} c(n) * x^(2*n) and d/dx (x*f(x)) = x^2 + f(x)^2. Now a(n) = - numerator of c(n+1) for n>=0. - Michael Somos, Apr 25 2020
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.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 74.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1, p. 19.
H. A. Rothe, in C. F. Hindenburg, editor, Sammlung Combinatorisch-Analytischer Abhandlungen, Vol. 2, Chap. XI. Fleischer, Leipzig, 1800, p. 331.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1958, p. 122, Exercise 2.
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..313 (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
a(n) = - numerator(A000182(n)/(4^n-1)) for n>0.
cot(x) = Sum_{k>=0} (-1)^k*B_{2*k}*4^k*x^(2*k-1)/(2*k)!.
a(n) = numerator(r(n)), with the negative rational numbers r(n) = [x^n]( (cot(sqrt(x))-1/sqrt(x))/sqrt(x)), 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
b := n -> (-1)^n*2^(2*n)*bernoulli(2*n)/(2*n)!;
a := n -> numer(b(n+1)); seq(a(i), i=-1..25);
# Peter Luschny, Jun 08 2009
MATHEMATICA
a[n_] := (-1)^(n+1)*4^(n+1)*BernoulliB[2*n+2]/(2*n+2)! // Numerator; Table[a[n], {n, -1, 25}] (* Jean-François Alcover, Apr 14 2014, after Peter Luschny *)
PROG
(PARI) apply(r->numerator(r), Vec(1/tan(x))) \\ Charles R Greathouse IV, Apr 14 2014
(PARI) a(n) = numerator((-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)).numerator()
[a(n) for n in range(-1, 25)] # Indranil Ghosh, Jun 23 2017
(Magma) [Numerator( (-1)^(n+1)*4^(n+1)*Bernoulli(2*n+2)/Factorial(2*n+2) ): n in [-1..25]]; // G. C. Greubel, Jul 03 2019
(Sage) [numerator( (-1)^(n+1)*4^(n+1)*bernoulli(2*n+2)/factorial(2*n+2) ) for n in (-1..25)] # G. C. Greubel, Jul 03 2019
(GAP) List([-1..25], n-> NumeratorRat( (-1)^(n+1)*4^(n+1)* Bernoulli(2*n+2)/Factorial(2*n+2) )) # G. C. Greubel, Jul 03 2019
CROSSREFS
KEYWORD
sign,frac,easy,nice
AUTHOR
STATUS
approved