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

%I A090681 #31 Sep 08 2022 08:45:12
%S A090681 1,1,31,5461,3202291,4722116521,14717667114151,86125672563201181,
%T A090681 868320396104950823611,14129659550745551130667441,
%U A090681 352552873457246307069012458671,12942188000689093683411117827763301,675618013651758631167025175564066787331,48743995308245045290420262686473639399176761
%N A090681 Expansion of (sec(x/2)^2 + sech(x/2)^2)/2 in powers of x^4.
%H A090681 Stefano Spezia, <a href="/A090681/b090681.txt">Table of n, a(n) for n = 0..135</a>
%F A090681 a(n) = -G(2n+1)/(2n+1) where G(k) is the k-th Genocchi number of first kind (A001469).
%F A090681 a(n) = A002425(2n+1).
%F A090681 a(n) = A012853(n)/2^(4n+1).
%F A090681 a(n) = abs(A012670(n)/2^(6n+1)).
%F A090681 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
%F A090681 a(n) == 1 (mod 30). - _Michael Somos_, Jul 23 2005
%e A090681 (sec(x/2)^2 + sech(x/2)^2)/2 = 1 + x^4/4! + 31*x^8/8! + 5461*x^12/12! + ...
%p A090681 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
%t A090681 a[n_]:=2*(2^(4*n+2)-1)*BernoulliB[4*n+2]/(2*n+1); Array[a,15,0] (* _Stefano Spezia_, Jun 14 2019 *)
%o A090681 (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 */
%o A090681 (PARI) a(n)=if(n<0,0,n=4*n+2;4*(2^n-1)*bernfrac(n)/n) /* _Michael Somos_, Mar 06 2004 */
%o A090681 (Magma) [2*(4^(2*n+1) -1)*BernoulliNumber(4*n+2)/(2*n+1): n in [0..15]]; // _G. C. Greubel_, Jun 28 2019
%o A090681 (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
%Y A090681 Cf. A001469, A002425, A012670, A012853.
%K A090681 nonn
%O A090681 0,3
%A A090681 _Benoit Cloitre_, Dec 18 2003