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 A002437 a(n) = A000364(n) * (3^(2*n+1) + 1)/4. (Formerly M4462 N1891) 12
 1, 7, 305, 33367, 6815585, 2237423527, 1077270776465, 715153093789687, 626055764653322945, 698774745485355051847, 968553361387420436695025, 1632180870878422847476890007, 3286322019402928956112227932705, 7791592461957309952817483706344167, 21485762937086358457367440231243675985 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The terms are multiples of the Euler numbers (A000364). REFERENCES 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. Glaisher, J. W. L.; Messenger of Math., 28 (1898), 36-79, see esp. p. 51. L. B. W. Jolley, Summation of Series, Dover, 2nd ed. (1961) 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). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, vol. 6, no. 1, #R21, (1999). FORMULA A000364(n) * (3^(2*n+1) + 1)/4. Q_2n(sqrt(3)), where the polynomials Q_n() are defined in A104035. - N. J. A. Sloane, Nov 06 2009 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 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 From Peter Bala, Feb 06 2017: (Start) G.f. 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) 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 EXAMPLE a(4) = A000364(4) * (3^(2*4+1)+1)/4 = 1385 * (3^9+1)/4 = 1385 * 4921 = 6815585. MAPLE 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; [seq(subs(u=sqrt(3), Q(2*n)), n=0..25)]; MATHEMATICA Table[Abs[EulerE[2 n]] (3^(2 n + 1) + 1) / 4, {n, 0, 30}] (* Vincenzo Librandi, Feb 07 2017 *) CROSSREFS Bisections: A156168, A156169. Cf. A000191, A001209, A012494, A000364, A000281, A156134. 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). Sequence in context: A015005 A209806 A257919 * A300870 A239163 A086215 Adjacent sequences:  A002434 A002435 A002436 * A002438 A002439 A002440 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Herman P. Robinson Further terms from N. J. A. Sloane, Nov 06 2009 STATUS approved

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Last modified May 26 09:37 EDT 2020. Contains 334620 sequences. (Running on oeis4.)