A328055 Expansion of e.g.f. -log(1 - x / (1 - x)^2).

0, 1, 5, 32, 270, 2904, 38400, 605520, 11113200, 232888320, 5488560000, 143704108800, 4138573824000, 130020673305600, 4425201196416000, 162194862064435200, 6369479157000960000, 266808274486161408000, 11874724379464826880000, 559591797303082672128000

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f.: log(1 + Sum_{k>=1} Fibonacci(2*k) * x^k).

a(n) = (n - 1)! * (Lucas(2*n) - 2) for n > 0.

Example

For n=2, the 3 labeled octupi are the following, and there are 2+2+1 ways to choose one element from each branch:
O-1-2;
O-2-1;
1-O-2. - Enrique Navarrete, Oct 29 2023

Mathematica

nmax = 19; CoefficientList[Series[-Log[1 - x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Join[{0}, Table[(n - 1)! (LucasL[2 n] - 2), {n, 1, 19}]]

Prog

(Magma) [0] cat [Factorial(n - 1)*(Lucas(2*n)-2):n in [1..20]]; // Marius A. Burtea, Oct 03 2019
(PARI) my(x='x+O('x^20)); concat(0, Vec(serlaplace(-log(1 - x / (1 - x)^2)))) \\ Michel Marcus, Oct 03 2019

Crossrefs

Cf. A001906, A004146, A005248, A005443, A029767, A052567 (exponential transform), A100404, A226968, A328054.

Sequence in context: A215916 A068102 A166993 * A265130 A305407 A320349

Adjacent sequences: A328052 A328053 A328054 * A328056 A328057 A328058

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 03 2019

Status

approved