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A080252 a(n) = n*a(n-1)+4*a(n-2)-4*(n-2)*a(n-3).

%I A080252 #19 Aug 23 2024 02:09:09
%S A080252 0,1,2,10,40,216,1296,9136,73088,658048,6580480,72386304,868635648,
%T A080252 11292267520,158091745280,2371376195584,37942019129344,
%U A080252 645014325264384,11610257854758912,220594899240681472,4411897984813629440
%N A080252 a(n) = n*a(n-1)+4*a(n-2)-4*(n-2)*a(n-3).
%F A080252 E.g.f.: sinh(z)*cosh(z)/(1-z).
%F A080252 a(n) ~ n!*(e^2-1/e^2)/4. - _Vaclav Kotesovec_, Oct 13 2012
%F A080252 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 A080252 := n -> (exp(2)*GAMMA(1+n,2) - exp(-2)*GAMMA(1+n,-2))/4:
%p A080252 seq(simplify(A080252(n)), n=0..20); # _Peter Luschny_, Dec 18 2017
%t A080252 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 A080252 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 A080252 (PARI) x='x+O('x^99); concat([0], Vec(serlaplace(sinh(x)*cosh(x)/(1-x)))) \\ _Altug Alkan_, Dec 18 2017
%Y A080252 Cf. A000023, A010842.
%K A080252 easy,nonn
%O A080252 0,3
%A A080252 Mario Catalani (mario.catalani(AT)unito.it), Feb 10 2003