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%I #18 May 06 2017 00:28:03
%S 1,1,13,421,26713,2794441,436186213,95033434861,27555582190513,
%T 10260037095841681,4771143086720391613,2710025439753915534901,
%U 1846296024220715321941513,1486014763274444231870834521
%N sec(arcsin(sinh(x)))=1+1/2!*x^2+13/4!*x^4+421/6!*x^6+26713/8!*x^8...
%F From _Michael Somos_, May 05 2017: (Start)
%F E.g.f.: Sum_{n>=0} a(n) * x^(2*n) / (2*n)! = sec(arcsin(sinh(x))) = 1 / sqrt(1 - sinh(x)^2).
%F E.g.f.: Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)! = F(i x| -1) / i where F(phi|m) is the elliptic integral of the 1st kind.
%F E.g.f. 1 / sqrt(1 - sinh(x)^2) = y satisfies 0 = y''*y + 2*y^2 - 3*y^4 - 3*y'^2 = y - 6*y^3 + 6*y^5 - y''.
%F a(n) = A012261(2*n). (End)
%e G.f. = 1 + x + 13*x^2 + 421*x^3 + 26713*x^4 + 2794441*x^5 + ...
%t a[ n_] := If[ n < 0, 0, With[ {m = 2 n + 1}, m! SeriesCoefficient[ EllipticF[ I x, -1] / I, {x, 0, m}]]]; (* _Michael Somos_, May 05 2017 *)
%t a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ 1 / Sqrt[1 - Sinh[x]^2], {x, 0, m}]]]; (* _Michael Somos_, May 05 2017 *)
%o (PARI) {a(n) = my(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / sqrt(1 - sinh(x + x * O(x^m))^2), m))}; /* _Michael Somos_, May 05 2017 */
%Y Cf. A012261.
%K nonn
%O 0,3
%A Patrick Demichel (patrick.demichel(AT)hp.com)
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