%I #26 Dec 16 2023 11:43:03

%S 0,1,20,310,4400,60100,806000,10711000,141680000,1869610000,

%T 24641000000,324555100000,4273412000000,56258281000000,

%U 740558540000000,9747925510000000,128308241600000000,1688851536100000000,22229288978000000000,292589141311000000000

%N a(n) = 20 * a(n-1) - 90 * a(n-2) for n>1, with a(0)=0, a(1)=1.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-90).

%F a(n) = ( (10 + sqrt(10))^n - (10 - sqrt(10))^n )/(2 * sqrt(10)).

%F a(n) = Sum_{k=0..floor((n-1)/2)} 10^(n-1-k) * binomial(n,2*k+1).

%F G.f.: x/(1 - 20 * x + 90 * x^2).

%F E.g.f.: exp(10 * x) * sinh(sqrt(10) * x) / sqrt(10).

%t LinearRecurrence[{20,-90},{0,1},20] (* _Harvey P. Dale_, Dec 16 2023 *)

%o (PARI) a(n) = polcoef(lift(Mod('x, ('x-10)^2-10)^n), 1);

%o (PARI) my(N=20, x='x+O('x^N)); concat (0, Vec(x/(1-20*x+90*x^2)))

%o (PARI) my(N=20, x='x+O('x^N)); concat (0, apply(round, Vec(serlaplace(exp(10*x)*sinh(sqrt(10)*x)/sqrt(10)))))

%Y Column k=10 of A361290.

%Y Cf. A289414.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Mar 12 2023