A081440 - OEIS

%I #14 Sep 08 2022 08:45:09

%S 1,1,2,4,20,76,632,3424,38096,265360,3682592,31332544,520705088,

%T 5232870592,101265169280,1173634791424,25911499036928,340187621683456,

%U 8436057652027904,123731966851240960,3404264757518332928

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

%C First binomial transform of expansion of cosh(x/sqrt(1-x^2)).

%H G. C. Greubel, <a href="/A081440/b081440.txt">Table of n, a(n) for n = 0..445</a>

%F D-finite with recurrence: a(n) = 2*a(n-1) + 3*(n-2)^2*a(n-2) - 3*(n-2)*(2*n-5)*a(n-3) - 3*(n-3)*(n-2)*(n^2 - 7*n + 11)*a(n-4) + 6*(n-4)^2*(n-3)*(n-2)*a(n-5) + (n-7)*(n-5)*(n-4)*(n-3)^2*(n-2)*a(n-6) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(2*n-11)*a(n-7) + (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*a(n-8). - _Vaclav Kotesovec_, Oct 29 2014

%p seq(coeff(series(exp(x)*cosh(x/sqrt(1-x^2)), x, n+1)*factorial(n), x, n), n = 0 .. 25); # _G. C. Greubel_, Aug 14 2019

%t With[{nn=25},CoefficientList[Series[Exp[x]Cosh[x/Sqrt[1-x^2]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jun 05 2014 *)

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace( exp(x)*cosh(x/sqrt(1-x^2)) )) \\ _G. C. Greubel_, Aug 14 2019

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x)*Cosh(x/Sqrt(1-x^2)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 14 2019

%o (Sage) [factorial(n)*( exp(x)*cosh(x/sqrt(1-x^2)) ).series(x,n+1).list()[n] for n in (0..25)] # _G. C. Greubel_, Aug 14 2019

%Y Cf. A081439, A081442.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Mar 21 2003

%E Definition clarified by _Harvey P. Dale_, Jun 05 2014