login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008990 Numerators of sequence having sqrt(cos(x)) as e.g.f. (even-indexed coefficients only). 1
1, -1, -1, -19, -559, -29161, -2368081, -276580459, -43947282079, -9118829535121, -2394495729300961, -776228170682260099, -304471093666800990799, -142128398853646068197881, -77865168574139358455774641, -49474260304294496117945326939 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

FORMULA

E.g.f.: sqrt(cos(sqrt(2)*x)) = 1 - x^2/2! - x^4/4! - 19x^6/6! -... - Ralf Stephan, Mar 03 2005

a(n) = sum(sum(binomial(k,j)*2^(n+2-2*k-j)*sum(binomial(j,i)*(j-2*i)^(2*n), i=0..floor((j-1)/2))*(-1)^(n+j+1), j=1..k)*C(k-1), k=1..2*n), n>0, C(n) - Catalan numbers (A000108). - Vladimir Kruchinin, Sep 10 2010

G.f.: 2/G(0) where G(k) =  2 - 4*x*(k+1)*(2*k-1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013

G.f.:  Q(0), where Q(k) = 1 - x*(k+1)*(2*k-1)/( x*(k+1)*(2*k-1) + 2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013

a(n) = 1/(4*n)*Sum_{k=1..n} C(2*n,2*k)*(-1)^(k)*2^(3*k)*(2^(2*k)-1)*B(2*k)*a(n-k), a(0)=1, where B(n) is Bernoulli numbers. - Vladimir Kruchinin, Jun 23 2015.

The odd terms of EllipticE(x,2) act as a g.f. for 2^n*a(n)/(2*n+1)!. - Benedict W. J. Irwin, Jun 06 2016

a(n) ~ -2^(5*n - 1) * n^(2*n - 1) / (Pi^(2*n - 1/2) * exp(2*n)). - Vaclav Kotesovec, Jun 11 2016

MATHEMATICA

n = 32; Partition[ CoefficientList[ Series[ Sqrt[Cos[Sqrt[2]*x]], {x, 0, n}], x]*Range[0, n]!, 2][[All, 1]] (* Jean-Fran├žois Alcover, Aug 30 2011 *)

Table[SeriesCoefficient[Series[EllipticE[x, 2], {x, 0, 41}], 2 n + 1] (2 n + 1)!/2^n, {n, 0, 20}] (* Benedict W. J. Irwin, Jun 06 2016 *)

PROG

(Maxima) C(n):=1/(n+1)*binomial(2*n, n); a(n):=sum(sum(binomial(k, j) *2^(n+2-2*k-j)*sum(binomial(j, i)*(j-2*i)^(2*n), i, 0, floor((j-1)/2))*(-1)^(n+j+1), j, 1, k)*C(k-1), k, 1, 2*n); /* Vladimir Kruchinin, Sep 10 2010 */

(Maxima) a(n):=if n=0 then 1 else 1/(4*n)*sum(binomial(2*n, 2*k)*(-1)^(k)*2^(3*k)*(2^(2*k)-1)*bern(2*k)*a(n-k), k, 1, n);  /* Vladimir Kruchinin, Jun 23 2015 */

CROSSREFS

Cf. A027641.

Denominators are in A000079.

Sequence in context: A278184 A035278 A092611 * A012845 A284111 A142023

Adjacent sequences:  A008987 A008988 A008989 * A008991 A008992 A008993

KEYWORD

sign,frac,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 19 16:55 EST 2018. Contains 299356 sequences. (Running on oeis4.)