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%I #11 Sep 08 2022 08:45:10
%S 1,3,9,29,97,327,1097,3649,12033,39371,127945,413349,1328609,4251535,
%T 13551753,43046729,136314625,430467219,1355971721,4261625389,
%U 13366006881,41841412823,130754415049,407953774929,1270932914177
%N Expansion of e.g.f. (1+x)*exp(2*x)*cosh(x).
%C Binomial transform of A082305 a(n)=(A006234(n)+A000027(n))/2
%H G. C. Greubel, <a href="/A082306/b082306.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-22,24,-9).
%F a(n) = (n + 1 + 3^(n-1)*(n + 3))/2.
%F G.f.: (1/(1-x)^2 + (1-2*x)/(1-3*x)^2)/2.
%F E.g.f.: (1+x)*exp(2*x)*cosh(x).
%t With[{nmax = 50}, CoefficientList[Series[(1 + x)*Exp[2*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Sep 16 2018 *)
%o (PARI) x='x+O('x^30); Vec(serlaplace((1+x)*exp(2*x)*cosh(x))) \\ _G. C. Greubel_, Sep 16 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(2*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Sep 16 2018
%Y Cf. A082307.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 09 2003