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A012109
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sec(arcsin(sinh(x)))=1+1/2!*x^2+13/4!*x^4+421/6!*x^6+26713/8!*x^8...
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1
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1, 1, 13, 421, 26713, 2794441, 436186213, 95033434861, 27555582190513, 10260037095841681, 4771143086720391613, 2710025439753915534901, 1846296024220715321941513, 1486014763274444231870834521
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OFFSET
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0,3
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LINKS
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Table of n, a(n) for n=0..13.
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FORMULA
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From Michael Somos, May 05 2017: (Start)
E.g.f.: Sum_{n>=0} a(n) * x^(2*n) / (2*n)! = sec(arcsin(sinh(x))) = 1 / sqrt(1 - sinh(x)^2).
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.
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''.
a(n) = A012261(2*n). (End)
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EXAMPLE
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G.f. = 1 + x + 13*x^2 + 421*x^3 + 26713*x^4 + 2794441*x^5 + ...
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MATHEMATICA
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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 *)
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 *)
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PROG
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(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 */
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CROSSREFS
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Cf. A012261.
Sequence in context: A081442 A100872 A012045 * A012084 A114759 A260871
Adjacent sequences: A012106 A012107 A012108 * A012110 A012111 A012112
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KEYWORD
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nonn
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AUTHOR
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Patrick Demichel (patrick.demichel(AT)hp.com)
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STATUS
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approved
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