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

0,3

Links

Table of n, a(n) for n=0..20.

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

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

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}]

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(x*(1+2*x)/(1-x-x^2)))) \\ Michel Marcus, Jun 07 2021

Crossrefs

Cf. A000204, A080833, A100404, A294222.

Sequence in context: A243273 A292502 A294361 * A204733 A195230 A316635

Adjacent sequences: A345072 A345073 A345074 * A345076 A345077 A345078

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 07 2021

Status

approved