A002431 Numerators in Taylor series for cot x. (Formerly M0124 N0050)

1, -1, -1, -2, -1, -2, -1382, -4, -3617, -87734, -349222, -310732, -472728182, -2631724, -13571120588, -13785346041608, -7709321041217, -303257395102, -52630543106106954746, -616840823966644, -522165436992898244102, -6080390575672283210764, -10121188937927645176372

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

Index entries for sequences related to Bernoulli numbers.

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

Cf. A000182, A036278 (denominators).

Sequence in context: A228005 A177438 A367762 * A259328 A202034 A323881

Adjacent sequences: A002428 A002429 A002430 * A002432 A002433 A002434

Keyword

sign,frac,easy,nice

Author

N. J. A. Sloane

Status

approved