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%I #18 Sep 08 2022 08:45:10
%S 0,1,2,10,32,116,392,1352,4608,15760,53792,183712,627200,2141504,
%T 7311488,24963200,85229568,290992384,993509888,3392055808,11581202432,
%U 39540700160,135000393728,460920178688,1573679923200,5372879343616
%N Binomial transform of sinh(x)*cosh(sqrt(2)*x).
%H G. C. Greubel, <a href="/A084154/b084154.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 (4,0,-8,4).
%F a(n) = 4*a(n-1) - 8*a(n-3) + 4*a(n-4), a(0)=0, a(1)=1, a(2)=2, a(3)=10.
%F a(n) = ((2+sqrt(2))^n + (2-sqrt(2))^n - sqrt(2)^n - (-sqrt(2))^n)/4.
%F G.f.: x*(1-2*x+2*x^2)/((1-2*x^2)*(1-4*x+2*x^2)).
%F E.g.f.: exp(x)*sinh(x)*cosh(sqrt(2)*x).
%t With[{nn=30},CoefficientList[Series[Exp[x]Sinh[x]Cosh[Sqrt[2]x],{x,0, nn}], x] Range[0,nn]!] (* or *) LinearRecurrence[{4,0,-8,4},{0,1,2, 10}, 30] (* _Harvey P. Dale_, Jun 19 2016 *)
%o (PARI) x='x+O('x^30); concat([0], Vec(x*(1-2*x+2*x^2)/((1-2*x^2)*(1-4*x+2*x^2)))) \\ _G. C. Greubel_, Aug 16 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-2*x+2*x^2)/((1-2*x^2)*(1-4*x+2*x^2)))); // _G. C. Greubel_, Aug 16 2018
%Y Cf. A084155.
%K easy,nonn
%O 0,3
%A _Paul Barry_, May 16 2003