A012853 Expansion of sec(x)^2+sech(x)^2 in powers of x^4.

2, 32, 15872, 44736512, 419730685952, 9902996106248192, 493842960380415967232, 46238368375619195682947072, 7458815407441059142195019251712, 1941965621570118224758799415904305152, 775271967544166063656029248004350271291392, 455362758259861772977200568672632329207841554432

Offset: 0

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

• nonn
• easy

Comments

log(sec(x)*cosh(x)) = 2/2!*x^2 + 32/6!*x^6 + 15872/10!*x^10...

Programs

PARI

{a(n) = if( n<0, 0, n=4*n + 2; 2 * (4^n - 2^n) * bernfrac(n) / n)} /* Michael Somos, Feb 14 2004 */

PARI

{a(n) = if( n<0, 0, n*=4; n! * polcoeff( cosh(x + x * O(x^n))^-2 + cos(x + x * O(x^n))^-2, n))} /* Michael Somos, Feb 05 2011*/

Formula

a(n) = (2^(2*k)) * (2^(2*k)-1) * B(2*k) / k with k = 2*n+1, where Bernoulli numbers B(2*k) = A027641(2*k) / A027642(2*k). That this gives integers follows from von Staudt's theorem. - Frank Ellermann, May 14 2001

E.g.f.: sec(x)^2 + sech(x)^2 = Sum_{k>=0} a(k) x^(4k) / (4k)!. - Michael Somos, Mar 06 2004

E.g.f.: log(cosh(x) / cos(x)) = Sum_{k>=0} a(k) x^(4k+2) / (4k+2)!. - Michael Somos, Feb 05 2011

a(n) = A001250(4*n) if n>0. - Michael Somos, Feb 05 2011

Extensions

Corrected by N. J. A. Sloane, Nov 07 2003