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A002431
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Numerators in Taylor series for cot x.
(Formerly M0124 N0050)
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4
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1, -1, -1, -2, -1, -2, -1382, -4, -3617, -87734, -349222, -310732, -472728182, -2631724, -13571120588, -13785346041608, -7709321041217, -303257395102, -52630543106106954746, -616840823966644, -522165436992898244102, -6080390575672283210764, -10121188937927645176372
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OFFSET
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-1,4
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COMMENTS
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Can be written as numerators of multiples of Bernoulli numbers.
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
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REFERENCES
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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.
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LINKS
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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].
Eric Weisstein's World of Mathematics, Cotangent
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FORMULA
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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
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EXAMPLE
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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).
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MAPLE
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b := n -> (-1)^n*2^(2*n)*bernoulli(2*n)/(2*n)!;
a := n -> numer(b(n+1)); seq(a(i), i=-1..25);
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MATHEMATICA
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PROG
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(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()
(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
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CROSSREFS
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KEYWORD
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sign,frac,easy,nice
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AUTHOR
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STATUS
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approved
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