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%I #16 Oct 29 2023 20:26:21
%S 0,1,5,32,270,2904,38400,605520,11113200,232888320,5488560000,
%T 143704108800,4138573824000,130020673305600,4425201196416000,
%U 162194862064435200,6369479157000960000,266808274486161408000,11874724379464826880000,559591797303082672128000
%N Expansion of e.g.f. -log(1 - x / (1 - x)^2).
%C a(n) is the number of ways to choose one element from each branch of labeled octupi with n nodes (cf. A029767 and example below). - _Enrique Navarrete_, Oct 29 2023
%F E.g.f.: log(1 + Sum_{k>=1} Fibonacci(2*k) * x^k).
%F a(n) = (n - 1)! * (Lucas(2*n) - 2) for n > 0.
%e For n=2, the 3 labeled octupi are the following, and there are 2+2+1 ways to choose one element from each branch:
%e O-1-2;
%e O-2-1;
%e 1-O-2. - _Enrique Navarrete_, Oct 29 2023
%t nmax = 19; CoefficientList[Series[-Log[1 - x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
%t Join[{0}, Table[(n - 1)! (LucasL[2 n] - 2), {n, 1, 19}]]
%o (Magma) [0] cat [Factorial(n - 1)*(Lucas(2*n)-2):n in [1..20]]; // _Marius A. Burtea_, Oct 03 2019
%o (PARI) my(x='x+O('x^20)); concat(0, Vec(serlaplace(-log(1 - x / (1 - x)^2)))) \\ _Michel Marcus_, Oct 03 2019
%Y Cf. A001906, A004146, A005248, A005443, A029767, A052567 (exponential transform), A100404, A226968, A328054.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Oct 03 2019
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