A246174 - OEIS

%I #6 Aug 19 2014 01:03:09

%S 2,1,3,3,8,8,4,1,15,20,15,5,30,48,44,24,6,1,56,105,119,84,35,7,104,

%T 224,296,256,144,48,8,1,189,459,696,711,495,228,63,9,340,920,1570,

%U 1840,1522,880,340,80,10,1,605,1804,3421,4521,4312,2981,1463,484,99,11

%N Triangle read by rows: T(n,k) is the number of vertex pairs at distance k of the Lucas cube Lambda(n) (1<=k<=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 number of entries in row n is equal to n if n is even and equal to n-1 if n is odd.

%C The entries in row n are the coefficients of the Hosoya polynomial of the Lucas cube Lambda(n).

%C T(n,1) = A099920(n-1) = number of edges in Lambda(n).

%C Sum(kT(n,k), k>=1) = A238420(n) = the Wiener index of Lambda(n).

%H S. Klavzar, M. Mollard, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match68/n1/match68n1_311-324.pdf">Wiener index and Hosoya polynomial of Fibonacci and Lucas cubes</a>, MATCH Commun. Math. Comput. Chem., 68, 2012, 311-324.

%H E. Munarini, C. P. Cippo, N. Z. Salvi, <a href="http://www.fq.math.ca/Scanned/39-1/munarini.pdf">On the Lucas cubes</a>, The Fibonacci Quarterly, 39, No. 1, 2001, 12-21.

%F G.f.: tz^2(2+t-z+tz-3tz^2+tz^3+tz^4)/((1+tz)(1-z-tz-z^2-tz^2+tz^3)(1-z-z^2)).  Derived from Theorem 4.3 of the Klavzar-Mollard reference in which the g.f. of the ordered Hosoya polynomials is given.

%e Row 2 is 2,1. Indeed, Lambda(2)  is the path-tree P(3) having vertex-pair distances 1,1, and 2.

%e Triangle starts:

%e 2,1;

%e 3,3;

%e 8,8,4,1;

%e 15,20,15,5;

%e 30,48,44,24,6,1;

%p g := t*z^2*(2+t-z+t*z-3*t*z^2+t*z^3+t*z^4)/((1+t*z)*(1-z-t*z-z^2-t*z^2+t*z^3)*(1-z-z^2)): gserz := simplify(series(g, z = 0, 20)): for j from 2 to 18 do H[j] := sort(coeff(gserz, z, j)) end do: for j from 2 to 13 do seq(coeff(H[j], t, k), k = 1 .. 2*floor((1/2)*j)) end do; # yields sequence in triangular form

%Y Cf. A099920, A238420, A246173.

%K nonn,tabf

%O 2,1

%A _Emeric Deutsch_, Aug 18 2014