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A086403 Numerators in continued fraction representation of (e-1)/(e+1).

%I A086403 #28 Apr 27 2016 03:51:15
%S A086403 1,6,61,860,15541,342762,8927353,268163352,9126481321,347074453550,
%T A086403 14586253530421,671314736852916,33580323096176221,1814008761930368850,
%U A086403 105246088515057569521,6527071496695499679152,430891964870418036393553,30168964612425958047227862
%N A086403 Numerators in continued fraction representation of (e-1)/(e+1).
%D A086403 Calvin C. Clawson, "Mathematical Mysteries", Perseus, 1999, p. 225.
%H A086403 Alois P. Heinz, <a href="/A086403/b086403.txt">Table of n, a(n) for n = 1..360</a>
%F A086403 Partial quotients in continued fraction representation of (e-1)/(e+1) are A016825: [2, 6, 10, 14, 18...], the convergents being: [2] = 1/2, [2, 6] = 6/13, [2, 6, 10] = 61/132...etc.; denominators are A079165 starting with n=1: 2, 13, 132, 1861, 33630, 741721, 19318376... 2. a(n) = closest integer to [(e-1)/(e+1)]*A079165(n), n>0
%F A086403 E.g.f.: sinh((1-sqrt(1-4*x))/2)/sqrt(1-4*x). - _Vladimir Kruchinin_, Apr 26 2016
%F A086403 a(n) = Sum_{k=1..(n+1)/2} (2*n-2*k+1)!/((2*k-1)!*(n-2*k+1)!). - _Vladimir Kruchinin_, Apr 26 2016
%F A086403 a(n) = -((-1)^n*sqrt(Pi/exp(1))*BesselI((2*n+1)/2, 1/2))/2 + (BesselK((2*n+1)/2, 1/2)*sinh(1/2))/sqrt(Pi), where BesselI(n,x) is the modified Bessel function of the first kind, BesselK(n,x) is the modified Bessel function of the second kind. - _Ilya Gutkovskiy_, Apr 26 2016
%F A086403 From _Vaclav Kotesovec_, Apr 27 2016: (Start)
%F A086403 a(n)/n! ~ BesselI(1/2, 1/2) * 2^(2*n-1)  / sqrt(n).
%F A086403 a(n) ~ sinh(1/2) * 2^(2*n + 1/2) * n^n  / exp(n).
%F A086403 (End)
%e A086403 a(4) = 860 = closest integer to[(e-1)/(e+1)]*A079165(4); = floor(860.0000292...) = 860. 860/1861 = [2, 6, 10, 14] = .462117141...; (e-1)/(e+1) = .462117157...
%p A086403 b:= proc(n) local i, q;
%p A086403       q:= 0;
%p A086403       for i to n do
%p A086403         q:= 1/(q+4*(n-i)+2)
%p A086403       od; q
%p A086403     end:
%p A086403 a:= n-> numer(b(n)):
%p A086403 seq(a(n), n=1..20);  # _Alois P. Heinz_, Feb 03 2012
%p A086403 numtheory:-cfrac((exp(1)-1)/(exp(1)+1),50,'convergents'):
%p A086403 map(numer,convergents[2..-2]); # _Robert Israel_, Apr 26 2016
%t A086403 Numerator@ FromContinuedFraction@ ContinuedFraction[(E - 1)/(E + 1), #] & /@ Range[2, 19] (* _Michael De Vlieger_, Apr 26 2016 *)
%o A086403 (Maxima)
%o A086403 a(n):=(sum((2*n-2*k+1)!/((2*k-1)!*(n-2*k+1)!),k,1,(n+1)/2));
%o A086403 taylor(sinh((1-sqrt(1-4*x))/2)/sqrt(1-4*x),x,0,10); /* _Vladimir Kruchinin_, Apr 26 2016 */
%Y A086403 Cf. A000108, A016825, A033184, A079165.
%K A086403 nonn
%O A086403 1,2
%A A086403 _Gary W. Adamson_, Jul 18 2003
%E A086403 More terms from _Alois P. Heinz_, Feb 03 2012