OFFSET
0,3
COMMENTS
Number of entries in row n is 1 + floor(n/2).
The entries in row n are the coefficients of the cube polynomial of the Lucas cube Lambda(n).
For n >= 1, sum of entries in row n = A014551(n) = 2^n + (-1)^n (the Jacobsthal-Lucas numbers).
Sum_{k >= 0} k*T(n,k) = A099429(n).
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).
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..10200 (rows 0..200 of the triangle, flattened).
Sandi Klavzar and Michel Mollard, Cube polynomial of Fibonacci and Lucas cubes, preprint.
Sandi Klavzar and Michel Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117, 2012, 93-105.
Jun Wan and Zuo-Ru Zhang, A proof of the only mode of a unimodal sequence, arXiv:2402.12858 [math.CO], 2024.
FORMULA
T(n,k) = Sum_{i = k..floor(n/2)} (2*binomial(n - i, i) - binomial(n - i - 1, i))*binomial(i, k).
G.f.: (1+(1+t)*z^2)/(1-z-(1+t)*z^2).
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.
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).
The generating function of column k (k >= 1) is z^(2k)(2-z)/(1-z-z^2)^(k+1).
EXAMPLE
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).
Triangle starts:
1;
1;
3, 2;
4, 3;
7, 8, 2;
11, 15, 5;
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 14 2014
STATUS
approved