A090681 Expansion of (sec(x/2)^2 + sech(x/2)^2)/2 in powers of x^4.

1, 1, 31, 5461, 3202291, 4722116521, 14717667114151, 86125672563201181, 868320396104950823611, 14129659550745551130667441, 352552873457246307069012458671, 12942188000689093683411117827763301, 675618013651758631167025175564066787331, 48743995308245045290420262686473639399176761

Offset: 0

Author

Benoit Cloitre, Dec 18 2003

Keywords

• nonn

Examples

			(sec(x/2)^2 + sech(x/2)^2)/2 = 1 + x^4/4! + 31*x^8/8! + 5461*x^12/12! + ...
		

Links

• Stefano Spezia, Table of n, a(n) for n = 0..135

Crossrefs

Cf. A001469, A002425, A012670, A012853.

Programs

Magma

[2*(4^(2*n+1) -1)*BernoulliNumber(4*n+2)/(2*n+1): n in [0..15]]; // G. C. Greubel, Jun 28 2019

Maple

a := n->(2*2^(4*n+2)-2)*bernoulli(4*n+2)/(2*n+1): seq(a(n), n = 0 .. 15); # Stefano Spezia, Jun 14 2019

Mathematica

a[n_]:=2*(2^(4*n+2)-1)*BernoulliB[4*n+2]/(2*n+1); Array[a,15,0] (* Stefano Spezia, Jun 14 2019 *)

PARI

a(n)=if(n<0,0,n*=4;n!*polcoeff(1/cosh(x/2+x*O(x^n))^2+1/cos(x/2+x*O(x^n))^2,n)/2) /* Michael Somos, Mar 06 2004 */

PARI

a(n)=if(n<0,0,n=4*n+2;4*(2^n-1)*bernfrac(n)/n) /* Michael Somos, Mar 06 2004 */

Sage

[2*(4^(2*n+1)-1)*bernoulli(4*n+2)/(2*n+1) for n in (0..15)] # G. C. Greubel, Jun 28 2019

Formula

a(n) = -G(2n+1)/(2n+1) where G(k) is the k-th Genocchi number of first kind (A001469).

a(n) = A002425(2n+1).

a(n) = A012853(n)/2^(4n+1).

a(n) = abs(A012670(n)/2^(6n+1)).

E.g.f.: (sec(x/2)^2 + sech(x/2)^2)/2 = Sum_{k>=1} a(k)*x^(4k)/(4k)!. - Michael Somos, Mar 06 2004

a(n) == 1 (mod 30). - Michael Somos, Jul 23 2005