A081442 - OEIS

%I #26 Sep 08 2022 08:45:09

%S 1,1,13,421,25369,2449801,346065061,67243537453,17192488230961,

%T 5593309059948049,2255588021494237501,1103994926592923677621,

%U 644587811150505183179593,442516027690815793746696601

%N Expansion of e.g.f.: cosh(x/sqrt(1-x^2)) (even powers).

%C Periodic zeros suppressed.

%H G. C. Greubel, <a href="/A081442/b081442.txt">Table of n, a(n) for n = 0..220</a>

%F a(n) = (2*n)!*Sum_{j=0..n} binomial(n-1,n-j)/(2*j)!. - _Vladimir Kruchinin_, May 19 2011

%F 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

%F 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

%F 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

%e cosh(x/sqrt(1-x^2)) = 1 + 1/2*x^2 + 13/24*x^4 + 421/720*x^6 + ...

%p 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

%t 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 *)

%o (Maxima) a(n):=(2*n)!*sum(binomial(n-1,n-j)/(2*j)!,j,0,n); /* _Vladimir Kruchinin_, May 19 2011 */

%o (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

%o (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

%o (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

%Y Cf. A081439, A081440.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Mar 21 2003

%E Definition corrected by _Joerg Arndt_, May 19 2011