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A081442
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Expansion of e.g.f.: cosh(x/sqrt(1-x^2)) (even powers).
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3
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1, 1, 13, 421, 25369, 2449801, 346065061, 67243537453, 17192488230961, 5593309059948049, 2255588021494237501, 1103994926592923677621, 644587811150505183179593, 442516027690815793746696601
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OFFSET
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0,3
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COMMENTS
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Periodic zeros suppressed.
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LINKS
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FORMULA
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E.g.f.: cosh(x/sqrt(1-x^2)) = 1 + x^2/(G(0)-x^2) where G(k)= 2*(2*k+1)*(k+1)*(1-x^2) + x^2 - 2*(2*k+1)*(k+1)*x^2*(1-x^2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 06 2012
D-finite with recurrence: a(n) = (12*n^2 - 24*n + 13)*a(n-1) - 12*(n-2)*(n-1)*(2*n-3)^2*a(n-2) + 16*(n-3)*(n-2)^2*(n-1)*(2*n-5)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Oct 29 2014
a(n) ~ 2^(2*n - 1/3) * n^(2*n - 1/3) * exp(3 * 2^(-2/3) * n^(1/3) - 2*n) / sqrt(3) * (1 - 19/72*2^(2/3)/n^(1/3) + 553/5184*2^(1/3)/n^(2/3)). - Vaclav Kotesovec, Oct 29 2014
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EXAMPLE
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cosh(x/sqrt(1-x^2)) = 1 + 1/2*x^2 + 13/24*x^4 + 421/720*x^6 + ...
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MAPLE
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seq(coeff(series(cosh(x/sqrt(1-x^2)), x, 2*n+1)*factorial(2*n), x, 2*n), n = 0 .. 20); # G. C. Greubel, Aug 14 2019
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MATHEMATICA
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Table[(CoefficientList[Series[Cosh[x/Sqrt[1-x^2]], {x, 0, 40}], x] * Range[0, 40]!)[[n]], {n, 1, 41, 2}] (* Vaclav Kotesovec, Oct 29 2014 *)
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PROG
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(Maxima) a(n):=(2*n)!*sum(binomial(n-1, n-j)/(2*j)!, j, 0, n); /* Vladimir Kruchinin, May 19 2011 */
(PARI) my(x='x+O('x^40)); v=Vec(serlaplace( cosh(x/sqrt(1-x^2)) )); vector(#v\2, n, v[2*n-1]) \\ G. C. Greubel, Aug 14 2019
(Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Cosh(x/Sqrt(1-x^2)) )); [Factorial(2*n-2)*b[2*n-1]: n in [1..Floor((m-2)/2)]]; // G. C. Greubel, Aug 14 2019
(Sage) [factorial(2*n)*( cosh(x/sqrt(1-x^2)) ).series(x, 2*n+1).list()[2*n] for n in (0..20)] # G. C. Greubel, Aug 14 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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