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A306823 a(n) is the second multiplicative Zagreb index of the Lucas cube Lambda(n). 1
1, 4, 27, 1048576, 45916502400000, 237376313799769806328950291431424, 18897697257047055734419223897702400000000000000000000000000000000000 (list; graph; refs; listen; history; text; internal format)
The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, and in which two vertices are adjacent when their Hamming distance is exactly 1.
The second multiplicative Zagreb index of a simple connected graph is product(deg(x)^(deg(x)) over all the vertices of the graph (see, for example, the I. Gutman reference, p. 16).
In the Maple program G = Sum_{n>=0} P[n]z^n is the generating function of the Lucas cubes according to size (coded by z) and vertex degrees (coded by t). See the Klavzar - Mollard - Petkovsek reference: l(x,y) on p. 1321 with different variables.
I. Gutman, Multiplicative Zagreb indices of trees, Bulletin of International Mathematical Virtual Institut ISSN 1840-4367, Vol. 1, 2011, 13-19.
S. Klavžar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.
a(2)=4 because the Lucas cube Lambda(2) is the path-tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently, a(2) = 1^1*1^1*2^2 = 4.
a(4)=1048576 because the Lucas cube Lambda(4) is a bouquet of tw 4-cycles, having 6 vertices of degree 2 and 1 vertex of degree 4; consequently, a(4) = (2^2)^6*4^4 = 2^12*v^4 = 1048576.
G:=(1+(1-t)*z+t^2*z^2+t*(1-t)*z^3-t*(1-t)^2*z^4)/((1-t*z)*(1-t*z^2)-t*z^3): Gser:=simplify(series(G, z=0, 50)): for n from 0 to 45 do P[n]:=sort(coeff(Gser, z, n)) od: seq(product(j^(j*coeff(P[n], t, j)), j=1..n), n=1..7);
Sequence in context: A000869 A132640 A307031 * A174212 A352838 A038706
Emeric Deutsch, Apr 16 2019

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