A080252 - OEIS

%I #19 Aug 23 2024 02:09:09

%S 0,1,2,10,40,216,1296,9136,73088,658048,6580480,72386304,868635648,

%T 11292267520,158091745280,2371376195584,37942019129344,

%U 645014325264384,11610257854758912,220594899240681472,4411897984813629440

%N a(n) = n*a(n-1)+4*a(n-2)-4*(n-2)*a(n-3).

%F E.g.f.: sinh(z)*cosh(z)/(1-z).

%F a(n) ~ n!*(e^2-1/e^2)/4. - _Vaclav Kotesovec_, Oct 13 2012

%F a(n) = (exp(2)*Gamma(1+n,2) - exp(-2)*Gamma(1+n,-2))/4 = (A010842(n) - A000023(n))/4. - _Peter Luschny_, Dec 18 2017

%p A080252 := n -> (exp(2)*GAMMA(1+n,2) - exp(-2)*GAMMA(1+n,-2))/4:

%p seq(simplify(A080252(n)), n=0..20); # _Peter Luschny_, Dec 18 2017

%t c = CoefficientList[Series[(Sinh[z]*Cosh[z])/(1 - z), {z, 0, 25}], z]; For[n = 0, n < 25, n++; Print[c[[n]]*(n - 1)! ]]

%t RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[n]==n*a[n-1]+4a[n-2]-4(n-2)* a[n-3]}, a,{n,20}] (* _Harvey P. Dale_, Nov 17 2013 *)

%o (PARI) x='x+O('x^99); concat([0], Vec(serlaplace(sinh(x)*cosh(x)/(1-x)))) \\ _Altug Alkan_, Dec 18 2017

%Y Cf. A000023, A010842.

%K easy,nonn

%O 0,3

%A Mario Catalani (mario.catalani(AT)unito.it), Feb 10 2003