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%I #25 Jan 09 2021 02:10:04
%S 1,9,9,1,1,18,63,68,23,2,1,27,162,350,310,114,15,1,1,36,306,996,1446,
%T 984,303,42,2,1,45,495,2155,4360,4360,2141,505,49,1,1,54,729,3976,
%U 10325,13650,9233,3124,468,20,1,63,1008,6608,20958,34482,29750,13170,2685,175,1,72,1332,10200,38220,75264,79002,43284,11190,980
%N Triangle read by rows: T(n,k) is the coefficient of (1+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 by 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 perfect matchings (i.e., Kekulé structures) with k proper sextets for the hexagonal graphene flake O(3,3,n).
%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 computed using ZZDecomposer (see link below), a graphical program to compute ZZ polynomials of benzenoids, or using ZZCalculator (see link below).
%H C.-P. Chou, <a href="https://bitbucket.org/solccp/zzdecomposer_binary/downloads/">ZZDecomposer executable</a>.
%H C.-P. Chou, <a href="https://github.com/solccp/zzcalculator">ZZCalculator source code</a>.
%H C.-P. Chou and H. A. Witek, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match68/n1/match68n1_3-30.pdf">An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids</a>, MATCH Commun. Math. Comput. Chem. 68 (2012), 3-30.
%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/S0012-365X(99)00293-9">The Clar covering polynomial of hexagonal systems III</a>, Discrete Math. 212 (2000), 261-269 (proper sextet is defined in Fig.1 and ZZ polynomial in the basis of (1+x)^k monomials is defined by Theorem 2).
%F T(n,k) = binomial(9,k)*binomial(n,k) + (10*binomial(7,k-2) - binomial(6,k-2))*binomial(n+1,k) + (20*binomial(5,k-4) + binomial(3,k-3) - binomial(3,k-5))*binomial(n+2,k) + (10*binomial(3,k-6) + binomial(2,k-5) + binomial(3,k-5))*binomial(n+3,k) + binomial(2,k-7)*binomial(n+4,k).
%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: 1 9 9 1
%e n=2: 1 18 63 68 23 2
%e n=3: 1 27 162 350 310 114 15 1
%e n=4: 1 36 306 996 1446 984 303 42 2
%e n=5: 1 45 495 2155 4360 4360 2141 505 49 1
%e n=6: 1 54 729 3976 10325 13650 9233 3124 468 20
%e n=7: 1 63 1008 6608 20958 34482 29750 13170 2685 175
%e n=8: 1 72 1332 10200 38220 75264 79002 43284 11190 980
%e ...
%e Row n=4 corresponds to the polynomial 1 + 36*(1+x) + 306*(1+x)^2 + 996*(1+x)^3 + 1446*(1+x)^4 + 984*(1+x)^5 + 303*(1+x)^6 + 42*(1+x)^7 + 2*(1+x)^8.
%p (n,k)->binomial(9,k)*binomial(n,k)+(10*binomial(7,k-2)-binomial(6,k-2))*binomial(n+1,k)+(20*binomial(5,k-4)+binomial(3,k-3)-binomial(3,k-5))*binomial(n+2,k)+(10*binomial(3,k-6)+binomial(2,k-5)+binomial(3,k-5))*binomial(n+3,k)+binomial(2,k-7)*binomial(n+4,k)
%Y Column k=0 is A000012.
%Y Column k=1 is A008591.
%Y Column k=2 is 9*A000566.
%Y Row sums give A047819.
%Y Row sums give column k=0 of A338158.
%Y Another representation is given by A338158.
%K nonn,tabf
%O 1,2
%A _Henryk A. Witek_, Oct 17 2020