A306823 - OEIS

%I #30 Mar 08 2020 23:14:45

%S 1,4,27,1048576,45916502400000,237376313799769806328950291431424,

%T 18897697257047055734419223897702400000000000000000000000000000000000

%N a(n) is the second multiplicative Zagreb index of the Lucas cube Lambda(n).

%C 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.

%C 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).

%C 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.

%H Alois P. Heinz, <a href="/A306823/b306823.txt">Table of n, a(n) for n = 1..11</a>

%H I. Gutman, <a href="http://www.imvibl.org/buletin/bulletin_1_13_19.pdf">Multiplicative Zagreb indices of trees</a>, Bulletin of International Mathematical Virtual Institut ISSN 1840-4367, Vol. 1, 2011, 13-19.

%H S. Klavžar, <a href="http://dx.doi.org/10.1007/s10878-011-9433-z">Structure of Fibonacci cubes: a survey</a>, J. Comb. Optim., 25, 2013, 505-522.

%H S. Klavžar, M. Mollard and M. Petkovšek, <a href="https://doi.org/10.1016/j.disc.2011.03.019">The degree sequence of Fibonacci and Lucas cubes</a>, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.

%e 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.

%e 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.

%p 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);

%Y Cf. A245960, A307580.

%K nonn

%O 1,2

%A _Emeric Deutsch_, Apr 16 2019