cp's OEIS Frontend

A307157 a(n) is the Narumi-Katayama index of the Fibonacci cube Gamma(n).

%I A307157 #16 Mar 27 2019 12:17:59
%S A307157 1,2,24,1152,1399680,290237644800,520105859481600000000,
%T A307157 3435834286784202670080000000000000000,
%U A307157 3045775242579858715944293498880000000000000000000000000000000000
%N A307157 a(n) is the Narumi-Katayama index of the Fibonacci cube Gamma(n).
%C A307157 The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and  in which two vertices are adjacent when their Hamming distance is exactly 1.
%C A307157 The Narumi-Katayama index of a connected graph is the product of the degrees of the vertices of the graph.
%H A307157 I. Gutman and M. Ghorbani, <a href="https://doi.org/10.1016/j.aml.2011.12.018">Some properties of the Narumi-Katayama index</a>, Applied Mathematics Letters, Vol. 25, No. 10 (2012), 1435-1438.
%H A307157 S. Klavžar, <a href="https://doi.org/10.1007/s10878-011-9433-z">Structure of Fibonacci cubes: a survey</a>, Journal of Combinatorial Optimization, Vol. 25, No. 4 (2013), 505-522.
%H A307157 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.
%F A307157 a(n) = Product_{k=1..n} k^T(n, k), where T(n, k) = Sum_{i=0..k} binomial(n-2*i, k-i)*binomial(i+1, n-k-i+1). T(n,k) gives the number of vertices of degree k in the Fibonacci cube Gamma(n) (see A245825).
%e A307157 a(2)=2 because the Fibonacci cube Gamma(2) is the path tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently, the Narumi-Katayama index is 1*1*2=2.
%p A307157 T := (n,k) -> add(binomial(n-2*i,k-i)*binomial(i+1,n-k-i+1), i=0..k):
%p A307157 seq(mul(j^T(n,j), j=1..n), n=1..10);
%Y A307157 Cf. A000045, A245825.
%K A307157 nonn,easy
%O A307157 1,2
%A A307157 _Emeric Deutsch_, Mar 27 2019