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%I #31 Jan 09 2021 02:10:00
%S 20,30,12,1,175,450,425,180,33,2,980,3308,4458,3065,1140,225,22,1,
%T 4116,16468,27293,24262,12521,3796,653,58,2,14112,63522,120848,126518,
%U 79506,30681,7132,933,58,1,41580,204180,429030,503664,361690,163380,45885,7588,648,20
%N Triangle read by rows: T(n,k) is the coefficient of x^k in the ZZ polynomial of the hexagonal graphene flake O(3,3,n).
%C The maximum k for which T(n,k) is nonzero, denoted as Cl(n), is usually referred to as the Clar number of O(3,3,n); one has: Cl(1)=3, Cl(2)=5, Cl(3)=7, Cl(4)=8, and Cl(n)=9 for n>4.
%C T(n,k) denotes the number of Clar covers of order k in the hexagonal graphene flake O(3,3,n).
%C The Kekulé number of O(3,3,n) is given by T(n, 0).
%C ZZ polynomials of hexagonal graphene flakes O(3,3,n) with n=1..10 are listed in Eq.(36) of Chou, Li and Witek.
%C ZZ polynomials of hexagonal graphene flakes O(3,3,n) with any n can be obtained from Eq.(13) of Witek, Langner, Mos and Chou.
%C ZZ polynomials of hexagonal graphene flakes O(3,3,n) can be also computed using ZZDecomposer (see links below), a graphical program to compute ZZ polynomials of general benzenoids.
%H C.-P. Chou, <a href="https://bitbucket.org/solccp/zzdecomposer_binary/downloads/">ZZDecomposer<\a>.
%H C.-P. Chou, Y. Li and H. A. Witek, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match68/n1/match68n1_31-64.pdf">Zhang-Zhang Polynomials of Various Classes of Benzenoid Systems</a>, MATCH Commun. Math. Comput. Chem. 68 (2012), 31-64.
%H C.-P. Chou and H. A. Witek, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match71/n3/match71n3_741-764.pdf">ZZDecomposer: A Graphical Toolkit for Analyzing the Zhang-Zhang Polynomials of Benzenoid Structures</a>, MATCH Commun. Math. Comput. Chem. 71 (2014), 741-764.
%H C.-P. Chou and H. A. Witek, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match72/n1/match72n1_75-104.pdf">Determination of Zhang-Zhang Polynomials for various Classes of Benzenoid Systems: Non-Heuristic Approach</a>, MATCH Commun. Math. Comput. Chem. 72 (2014), 75-104.
%H S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 105 for a graphical definition of O(3,3,n)).
%H H. A. Witek, J. Langner, G. Mos, and C.-P. Chou, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match78/n2/match78n2_487-504.pdf">Zhang-Zhang Polynomials of Regular 5-tier Benzeonid Strips</a>, MATCH Commun. Math. Comput. Chem. 78 (2017), 487-504.
%H H. Zhang and F. Zhang, <a href="https://doi.org/10.1016/0166-218X(95)00081-2">The Clar covering polynomial of hexagonal systems I</a>, Discrete Appl. Math. 69 (1996), 147-167 (ZZ polynomial is defined by Eq.(2.1) and working formula is given by Eq.(2.2)).
%F T(n,k) = Sum_{l=0..9} (binomial(k+l,k)*(binomial(9,k+l)*binomial(n,k+l)+(10*binomial(7,k+l-2)-binomial(6,k+l-2))*binomial(n+1,k+l)+(20*binomial(5,k+l-4)+binomial(3,k+l-3)-binomial(3,k+l-5))*binomial(n+2,k+l)+(10*binomial(3,k+l-6)+binomial(2,k+l-5)+binomial(3,k+l-5))*binomial(n+3,k+l)+binomial(2,k+l-7)*binomial(n+4,k+l)).
%F This formula can be obtained by a double sum rotation from Eq.(13) of Witek, Langner, Mos and Chou. Eq.(13) was first discovered heuristically as Eq.(37) of Chou, Li and Witek; a formal proof was given in Eqs.(66-71) on pp. 100-102 of Chou and Witek.
%e Triangle begins:
%e k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9
%e n=1: 20 30 12 1
%e n=2: 175 450 425 180 33 2
%e n=3: 980 3308 4458 3065 1140 225 22 1
%e n=4: 4116 16468 27293 24262 12521 3796 653 58 2
%e n=5: 14112 63522 120848 126518 79506 30681 7132 933 58 1
%e n=6: 41580 204180 429030 503664 361690 163380 45885 7588 648 20
%e ...
%e Row n=4 corresponds to the polynomial 4116 + 16468*x + 27293*x^2 + 24262*x^3 + 12521*x^4 + 3796*x^5 + 653*x^6 + 58*x^7 + 2*x^8.
%p (n,k)->add(binomial(i+k,k)*(binomial(9,i+k)*binomial(n,i+k)+(10*binomial(7,i+k-2)-binomial(6,i+k-2))*binomial(n+1,i+k)+(20*binomial(5,i+k-4)+binomial(3,i+k-3)-binomial(3,i+k-5))*binomial(n+2,i+k)+(10*binomial(3,i+k-6)+binomial(2,i+k-5)+binomial(3,i+k-5))*binomial(n+3,i+k)+binomial(2,i+k-7)*binomial(n+4,i+k)),i = 0..9)
%Y Column k=0 is A047819.
%Y Other representation of ZZ polynomials of O(3,3,n) is given by A338217.
%K nonn,tabf
%O 1,1
%A _Henryk A. Witek_, Oct 14 2020
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