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%I #24 Jan 09 2021 02:10:10
%S 35,60,30,4,490,1470,1695,940,255,30,1,4116,16468,27293,24262,12521,
%T 3796,653,58,2,24696,118590,243994,281372,199822,90482,26195,4748,517,
%U 32,10,116424,635362,1513660,2068248,1791158,1025836,393659,100450,16583,1678,930,21
%N Triangle read by rows: T(n,k) is the coefficient of x^k in the ZZ polynomial of the hexagonal graphene flake O(3,4,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(3,4,n); one has: Cl(1)=3, Cl(2)=6, Cl(3)=8, Cl(4)=10, Cl(5)=11, and Cl(n)=12 for n>5.
%C T(n,k) denotes the number of Clar covers of order k in the hexagonal graphene flake O(3,4,n).
%C The Kekulé number of O(3,4,n) is given by T(n, 0).
%C ZZ polynomials of hexagonal graphene flakes O(3,4,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 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 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(3,4,n)).
%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_{i=0..12} binomial(k+i,k)*(binomial(n,k+i)*binomial(12,k+i) + 18*binomial(n+1,k+i)*binomial(10,k+i-2) + 84*binomial(n+2,k+i)*binomial(8,k+i-4) + 126*binomial(n+3,k+i)*binomial(6,k+i-6) + 57*binomial(n+4,k+i)*binomial(4,k+i-8) + 4*binomial(n+5,k+i)*binomial(2,k+i-10) + Sum_{h=0..1} (4*binomial(n+1+h,k+i)*binomial(9,k+i-3) + 24*binomial(n+2+h,k+i)*binomial(7,k+i-5) + 36*binomial(n+3+h,k+i)*binomial(5,k+i-7) + 14*binomial(n+4+h,k+i)*binomial(3,k+i-9)) + Sum_{s=0..2} Sum_{h=0..2} binomial(2,s)*binomial(2,h)*binomial(n+2+s+h,k+i)*binomial(6-2*s,k+i-6-2*s)) (conjectured, explicitly confirmed for n=1..1000).
%e Triangle begins:
%e k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k=10
%e n=1: 35 60 30 4
%e n=2: 490 1470 1695 940 255 30 1
%e n=3: 4116 16468 27293 24262 12521 3796 653 58 2
%e n=4: 24696 118590 243994 281372 199822 90482 26195 4748 517 32 1
%e n=5: 116424 635362 1513660 2068248 1791158 1025836 393659 100450 16583 1678 93 2
%e ...
%e Row n=4 corresponds to the polynomial 24696 + 118590*x + 243994*x^2 + 281372*x^3 + 199822*x^4 + 90482*x^5 + 26195*x^6 + 4748*x^7 + 517*x^8 + 32*x^9 + x^10.
%p (n,k)->add(binomial(k+i,k)*(binomial(n,k+i)*binomial(12,k+i)+18*binomial(n+1,k+i)*binomial(10,k+i-2)+84*binomial(n+2,k+i)*binomial(8,k+i-4)+126*binomial(n+3,k+i)*binomial(6,k+i-6)+57*binomial(n+4,k+i)*binomial(4,k+i-8)+4*binomial(n+5,k+i)*binomial(2,k+i-10)+add(4*binomial(n+1+h,k+i)*binomial(9,k+i-3)+24*binomial(n+2+h,k+i)*binomial(7,k+i-5)+36*binomial(n+3+h,k+i)*binomial(5,k+i-7)+14*binomial(n+4+h,k+i)*binomial(3,k+i-9),h = 0 .. 1)+add(add(binomial(2,s)*binomial(2,h)*binomial(n+2+s+h,k+i)*binomial(6-2*s,k+i-6-2*s),s = 0 .. 2),h = 0 .. 2)),i = 0 .. 12).
%Y Column k=0 is A107915.
%K nonn,tabf
%O 1,1
%A _Henryk A. Witek_, Oct 18 2020