A002437 a(n) = A000364(n) * (3^(2*n+1) + 1)/4. (Formerly M4462 N1891)

1, 7, 305, 33367, 6815585, 2237423527, 1077270776465, 715153093789687, 626055764653322945, 698774745485355051847, 968553361387420436695025, 1632180870878422847476890007, 3286322019402928956112227932705, 7791592461957309952817483706344167, 21485762937086358457367440231243675985

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.

J. W. L. Glaisher, 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)

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)

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, A349429.

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 * A376464 A300870 A239163

Adjacent sequences: A002434 A002435 A002436 * A002438 A002439 A002440

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Herman P. Robinson

Further terms from N. J. A. Sloane, Nov 06 2009

Status

approved