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A245962 Triangle read by rows: T(n,k) is the number of induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the hypercube Q(k).

%I A245962 #33 Mar 01 2024 07:20:15
%S A245962 1,1,3,2,4,3,7,8,2,11,15,5,18,30,15,2,29,56,35,7,47,104,80,24,2,76,
%T A245962 189,171,66,9,123,340,355,170,35,2,199,605,715,407,110,11,322,1068,
%U A245962 1410,932,315,48,2,521,1872,2730,2054,832,169,13,843,3262,5208,4396,2079,532,63,2
%N A245962 Triangle read by rows: T(n,k) is the number of induced subgraphs of the Lucas cube Lambda(n) that are isomorphic to the hypercube Q(k).
%C A245962 Number of entries in row n is 1 + floor(n/2).
%C A245962 The entries in row n are the coefficients of the cube polynomial of the Lucas cube Lambda(n).
%C A245962 For n >= 1, sum of entries in row n = A014551(n) = 2^n + (-1)^n (the Jacobsthal-Lucas numbers).
%C A245962 Sum_{k >= 0} k*T(n,k) = A099429(n).
%C A245962 T(n,0) = A000032(n) (n >= 1; the Lucas numbers); T(n,1) = A099920(n-1); T(n,2) = A245961(n).
%C A245962 As communicated by the authors, Theorem 5.2 and Corollary 5.3 of the Klavzar et al. paper contains a typo: 2nd binomial should be binomial(n - a - 1, a) resp. binomial(n - i - 1, i).
%H A245962 Paolo Xausa, <a href="/A245962/b245962.txt">Table of n, a(n) for n = 0..10200</a> (rows 0..200 of the triangle, flattened).
%H A245962 Sandi Klavzar and Michel Mollard, <a href="http://www-fourier.ujf-grenoble.fr/~mollard/accepte/CubePolyRevised.pdf">Cube polynomial of Fibonacci and Lucas cubes</a>, preprint.
%H A245962 Sandi Klavzar and Michel Mollard, <a href="http://dx.doi.org/10.1007/s10440-011-9652-4">Cube polynomial of Fibonacci and Lucas cubes</a>, Acta Appl. Math. 117, 2012, 93-105.
%H A245962 Jun Wan and Zuo-Ru Zhang, <a href="https://arxiv.org/abs/2402.12858">A proof of the only mode of a unimodal sequence</a>, arXiv:2402.12858 [math.CO], 2024.
%F A245962 T(n,k) = Sum_{i = k..floor(n/2)} (2*binomial(n - i, i) - binomial(n - i - 1, i))*binomial(i, k).
%F A245962 G.f.: (1+(1+t)*z^2)/(1-z-(1+t)*z^2).
%F A245962 The generating polynomial of row n (i.e., the cube polynomial of Lambda(n)) is Sum_{i = 0..floor(n/2)} (2*binomial(n - i, i) - binomial(n - i - 1))(1+x)^i.
%F A245962 The generating polynomial of row n (i.e., the cube polynomial of Lambda(n)) is ((1+w)/2)^n + ((1-w)/2)^n, where w = sqrt(5 + 4x).
%F A245962 The generating function of column k (k >= 1) is z^(2k)(2-z)/(1-z-z^2)^(k+1).
%e A245962 Row 4 is 7, 8, 2. Indeed, the Lucas cube Lambda(4) is the graph <><> obtained by identifying a vertex of a square with a vertex of another square; it has 7 vertices (i.e., Q(0)s), 8 edges (i.e., Q(1)s), and 2 squares (i.e., Q(2)s).
%e A245962 Triangle starts:
%e A245962    1;
%e A245962    1;
%e A245962    3,  2;
%e A245962    4,  3;
%e A245962    7,  8,  2;
%e A245962   11, 15,  5;
%p A245962 T := proc (n, k) options operator, arrow: add((2*binomial(n-i, i)-binomial(n-i-1, i))*binomial(i, k), i = k .. floor((1/2)*n)) end proc: for n from 0 to 20 do seq(T(n, k), k = 0 .. (1/2)*n) end do; # yields sequence in triangular form
%t A245962 A245962[n_, k_] := Sum[(2*Binomial[n-i, i]-Binomial[n-i-1, i])*Binomial[i, k], {i, k, n/2}]; Table[A245962[n, k], {n, 0, 15}, {k, 0, n/2}] (* _Paolo Xausa_, Feb 29 2024 *)
%Y A245962 Cf. A000032, A014551, A099429, A099920, A245961.
%K A245962 nonn,tabf
%O A245962 0,3
%A A245962 _Emeric Deutsch_, Aug 14 2014