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%I #14 Jan 03 2025 03:15:13
%S 0,1,1,1,1,1,691,2,3617,43867,174611,155366,236364091,1315862,
%T 3392780147,6892673020804,7709321041217,151628697551,
%U 26315271553053477373,308420411983322,261082718496449122051,3040195287836141605382,2530297234481911294093
%N Numerators of coefficients at even powers in Taylor series expansion of log(x/sin(x)).
%C This sequence shares many terms with A046988 (and appears to have been erroneously confused with it), but actually differs from it at indexes 0, 14, 22, 26, 28, 30, 38, 42, 44, 46, 50, 52, 54, 56, 58, 60, ...
%D L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205
%D T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 222, series for log(H(x)/x).
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
%D Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:4 at page 301.
%F log(x/sin(x)) = Sum_{n>0} (2^(2*n-1)*(-1)^(n+1)*B(2*n)/(n*(2*n)!) * x^(2*n)). - _Ralf Stephan_, Apr 01 2015 [corrected by _Roland J. Etienne_, Apr 19 2016]
%e log(x/sin(x)) = (1/6)*x^2 + (1/180)*x^4 + (1/2835)*x^6 + (1/37800)*x^8 + (1/467775)*x^10 + (691/3831077250)*x^12 + ...
%t a[0] = 0; a[n_] := Numerator[((-1)^(n + 1) 2^(2 n - 1) BernoulliB[2 n])/(n (2 n)!)]; Table[a[n], {n, 0, 20}] (* or *)
%t Numerator@Table[SeriesCoefficient[Log[x/Sin[x]], {x, 0, 2n}], {n, 0, 20}]
%Y Cf. A046989 (denominators), A046988.
%K nonn,frac,nice
%O 0,7
%A _Vladimir Reshetnikov_, Mar 04 2017