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A081442
Expansion of e.g.f.: cosh(x/sqrt(1-x^2)) (even powers).
3
1, 1, 13, 421, 25369, 2449801, 346065061, 67243537453, 17192488230961, 5593309059948049, 2255588021494237501, 1103994926592923677621, 644587811150505183179593, 442516027690815793746696601
OFFSET
0,3
COMMENTS
Periodic zeros suppressed.
LINKS
FORMULA
a(n) = (2*n)!*Sum_{j=0..n} binomial(n-1,n-j)/(2*j)!. - Vladimir Kruchinin, May 19 2011
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
EXAMPLE
cosh(x/sqrt(1-x^2)) = 1 + 1/2*x^2 + 13/24*x^4 + 421/720*x^6 + ...
MAPLE
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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Sequence in context: A308341 A098890 A012023 * A100872 A012045 A012109
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 21 2003
EXTENSIONS
Definition corrected by Joerg Arndt, May 19 2011
STATUS
approved