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A246174 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).

%I A246174 #8 Jan 05 2025 19:51:40
%S A246174 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 A246174 224,296,256,144,48,8,1,189,459,696,711,495,228,63,9,340,920,1570,
%U A246174 1840,1522,880,340,80,10,1,605,1804,3421,4521,4312,2981,1463,484,99,11
%N A246174 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 A246174 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 A246174 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 A246174 The entries in row n are the coefficients of the Hosoya polynomial of the Lucas cube Lambda(n).
%C A246174 T(n,1) = A099920(n-1) = number of edges in Lambda(n).
%C A246174 Sum(kT(n,k), k>=1) = A238420(n) = the Wiener index of Lambda(n).
%H A246174 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 A246174 E. Munarini, C. P. Cippo, N. Z. Salvi, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-1/munarini.pdf">On the Lucas cubes</a>, The Fibonacci Quarterly, 39, No. 1, 2001, 12-21.
%F A246174 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 A246174 Row 2 is 2,1. Indeed, Lambda(2)  is the path-tree P(3) having vertex-pair distances 1,1, and 2.
%e A246174 Triangle starts:
%e A246174 2,1;
%e A246174 3,3;
%e A246174 8,8,4,1;
%e A246174 15,20,15,5;
%e A246174 30,48,44,24,6,1;
%p A246174 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 A246174 Cf. A099920, A238420, A246173.
%K A246174 nonn,tabf
%O A246174 2,1
%A A246174 _Emeric Deutsch_, Aug 18 2014