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%I #13 Aug 20 2021 05:14:22
%S 1,1,7,43,409,4441,58351,872467,14776273,278033329,5759752951,
%T 130094213371,3181051122217,83674165333513,2355245699211679,
%U 70617410638402531,2246412316372784161,75551901666095113057,2678119105038094325863,99778611508176786458059,3897493112463397722989881
%N E.g.f.: exp( x*(1 + 2*x) / (1 - x - x^2) ).
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * k! * Lucas(k) * a(n-k).
%F a(n) ~ (1 + sqrt(5))^n * exp(1/(2*sqrt(5)) - 1 + 2*sqrt(n) - n) * n^(n - 1/4) / 2^(n + 1/2). - _Vaclav Kotesovec_, Jun 08 2021
%F D-finite with recurrence a(n) +(-2*n+1)*a(n-1) -(n+2)*(n-1)*a(n-2) +(2*n-5)*(n-1)*(n-2)*a(n-3) +(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Aug 20 2021
%p A345075 := proc(n)
%p option remember ;
%p if n = 0 then
%p 1;
%p else
%p add(binomial(n-1,k-1)*k!*procname(n-k)*A000204(k),k=1..n) ;
%p end if;
%p end proc:
%p seq(A345075(n),n=0..42) ; # _R. J. Mathar_, Aug 20 2021
%t nmax = 20; CoefficientList[Series[Exp[x (1 + 2 x)/(1 - x - x^2)], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] k! LucasL[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(x*(1+2*x)/(1-x-x^2)))) \\ _Michel Marcus_, Jun 07 2021
%Y Cf. A000204, A080833, A100404, A294222.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jun 07 2021
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