|
%I #16 Apr 23 2014 02:02:50
%S 2,32,15872,44736512,419730685952,9902996106248192,
%T 493842960380415967232,46238368375619195682947072,
%U 7458815407441059142195019251712,1941965621570118224758799415904305152,775271967544166063656029248004350271291392,455362758259861772977200568672632329207841554432
%N Expansion of sec(x)^2+sech(x)^2 in powers of x^4.
%C log(sec(x)*cosh(x)) = 2/2!*x^2 + 32/6!*x^6 + 15872/10!*x^10...
%F 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
%F E.g.f.: sec(x)^2 + sech(x)^2 = Sum_{k>=0} a(k) x^(4k) / (4k)!. - _Michael Somos_, Mar 06 2004
%F E.g.f.: log(cosh(x) / cos(x)) = Sum_{k>=0} a(k) x^(4k+2) / (4k+2)!. - _Michael Somos_, Feb 05 2011
%F a(n) = A001250(4*n) if n>0. - _Michael Somos_, Feb 05 2011
%o (PARI) {a(n) = if( n<0, 0, n=4*n + 2; 2 * (4^n - 2^n) * bernfrac(n) / n)} /* _Michael Somos_, Feb 14 2004 */
%o (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*/
%K nonn,easy
%O 0,1
%A Patrick Demichel (patrick.demichel(AT)hp.com)
%E Corrected by _N. J. A. Sloane_, Nov 07 2003
| 