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%I M4462 N1891 #52 Apr 21 2022 13:53:03
%S 1,7,305,33367,6815585,2237423527,1077270776465,715153093789687,
%T 626055764653322945,698774745485355051847,968553361387420436695025,
%U 1632180870878422847476890007,3286322019402928956112227932705,7791592461957309952817483706344167,21485762937086358457367440231243675985
%N a(n) = A000364(n) * (3^(2*n+1) + 1)/4.
%C The terms are multiples of the Euler numbers (A000364).
%D 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. 75.
%D J. W. L. Glaisher, Messenger of Math., 28 (1898), 36-79, see esp. p. 51.
%D L. B. W. Jolley, Summation of Series, Dover, 2nd ed. (1961)
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002437/b002437.txt">Table of n, a(n) for n = 0..200</a>
%H Michael E. Hoffman, <a href="https://doi.org/10.37236/1453">Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences</a>, The Electronic Journal of Combinatorics, vol. 6, no. 1, #R21, (1999).
%F A000364(n) * (3^(2*n+1) + 1)/4.
%F Q_2n(sqrt(3)), where the polynomials Q_n() are defined in A104035. - _N. J. A. Sloane_, Nov 06 2009
%F a(n) = (-1)^n*Sum_{k = 0..2*n-1} w^(2*n+k)*Sum_{j = 1..2*n-1} (-1)^(k-j)*binomial(2*n-1,k-j)*(2*j - 1)^(2*n-2), where w = exp(2*Pi*i/6) (i = sqrt(-1)). Cf. A002439. - _Peter Bala_, Jan 21 2011
%F Sum_{n>=1} (-1)^floor((n-1)/2) 1/A007310(n)^s = r_s with r_{2s+1} = 2 *(Pi/6)^(2s+1) *a(s) /(2s)!. [Jolley eq (315)]. - _R. J. Mathar_, Mar 24 2011
%F From _Peter Bala_, Feb 06 2017: (Start)
%F E.g.f.: cos(x)^2/cos(3*x) = cos(x)/(1 - 4*sin(x)^2) = 1 + 7*x^2/2! + 305*x^4/4! + 33367*x^6/6! + .... This is the even part of (1/2)*sec(x + Pi/3). Cf. A000191. (End)
%F a(n) = (1/2)*Integral_{x = 0..inf} x^(2*n)*cosh(Pi*x/3)/cosh(Pi*x/2) dx. - Cf. A000281. - _Peter Bala_, Nov 08 2019
%e a(4) = A000364(4) * (3^(2*4+1)+1)/4 = 1385 * (3^9+1)/4 = 1385 * 4921 = 6815585.
%p Q:=proc(n) option remember; if n=0 then RETURN(1); else RETURN(expand((u^2+1)*diff(Q(n-1),u)+u*Q(n-1))); fi; end;
%p [seq(subs(u=sqrt(3),Q(2*n)),n=0..25)];
%t Table[Abs[EulerE[2 n]] (3^(2 n + 1) + 1) / 4, {n, 0, 30}] (* _Vincenzo Librandi_, Feb 07 2017 *)
%Y Bisections: A156168, A156169.
%Y Cf. A000191, A001209, A012494, A000364, A000281, A156134, A349429.
%Y Cf. other sequences with a g.f. of the form cos(x)/(1 - k*sin^2(x)): A012494 (k=-1), A001209 (k=1/2), A000364 (k=1), A000281 (k=2), A156134 (k=3).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Herman P. Robinson_
%E Further terms from _N. J. A. Sloane_, Nov 06 2009
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