A345075 E.g.f.: exp( x*(1 + 2*x) / (1 - x - x^2) ).
1, 1, 7, 43, 409, 4441, 58351, 872467, 14776273, 278033329, 5759752951, 130094213371, 3181051122217, 83674165333513, 2355245699211679, 70617410638402531, 2246412316372784161, 75551901666095113057, 2678119105038094325863, 99778611508176786458059, 3897493112463397722989881
Offset: 0
Keywords
Programs
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Maple
A345075 := proc(n) option remember ; if n = 0 then 1; else add(binomial(n-1,k-1)*k!*procname(n-k)*A000204(k),k=1..n) ; end if; end proc: seq(A345075(n),n=0..42) ; # R. J. Mathar, Aug 20 2021
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Mathematica
nmax = 20; CoefficientList[Series[Exp[x (1 + 2 x)/(1 - x - x^2)], {x, 0, nmax}], x] Range[0, nmax]! 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}] -
PARI
my(x='x+O('x^30)); Vec(serlaplace(exp(x*(1+2*x)/(1-x-x^2)))) \\ Michel Marcus, Jun 07 2021
Formula
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * k! * Lucas(k) * a(n-k).
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
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