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A338259 Triangle read by rows: T(n,k) is the coefficient of (1+x)^k in the ZZ polynomial of the hexagonal graphene flake O(3,4,n).

%I A338259 #17 Jan 09 2021 02:10:40
%S A338259 1,12,18,41,24,120,200,120,24,11,36,306,996,1446,984,303,42,21,48,576,
%T A338259 2800,6525,7848,4957,1644,274,22,11,60,930,6020,19365,33600,32487,
%U A338259 17694,5336,858,71,21,72,1368,11064,45435,103200,134806,102912,45567,11358,1510,86,1
%N A338259 Triangle read by rows: T(n,k) is the coefficient of (1+x)^k in the ZZ polynomial of the hexagonal graphene flake O(3,4,n).
%C A338259 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 A338259 T(n,k) denotes the number of perfect matchings (i.e., Kekulé structures) with k proper sextets for the hexagonal graphene flake O(3,4,n).
%C A338259 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 A338259 C.-P. Chou, <a href="https://bitbucket.org/solccp/zzdecomposer_binary/downloads/">ZZDecomposer executable</a>.
%H A338259 C.-P. Chou, <a href="https://github.com/solccp/zzcalculator">ZZCalculator source code</a>.
%H A338259 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 A338259 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 A338259 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 A338259 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 A338259 T(n,k) = binomial(n,k)*binomial(12,k) + 18*binomial(n+1,k)*binomial(10,k-2) + 84*binomial(n+2,k)*binomial(8,k-4) + 126*binomial(n+3,k)*binomial(6,k-6) + 57*binomial(n+4,k)*binomial(4,k-8) + 4*binomial(n+5,k)*binomial(2,k-10) + Sum_{h=0..1} (4*binomial(n+1+h,k)*binomial(9,k-3) + 24*binomial(n+2+h,k)*binomial(7,k-5) + 36*binomial(n+3+h,k)*binomial(5,k-7) + 14*binomial(n+4+h,k)*binomial(3,k-9)) + Sum_{s=0..2} Sum_{h=0..2} binomial(2,s)*binomial(2,h)*binomial(n+2+s+h,k)*binomial(6-2*s,k-6-2*s) (conjectured, explicitly confirmed for n=1..1000).
%e A338259 Triangle begins:
%e A338259    k=0 k=1 k=2   k=3    k=4    k=5    k=6    k=7   k=8   k=9 k=10 k=11 k=12
%e A338259 n=1: 1 12   18     4
%e A338259 n=2: 1 24  120   200    120     24      1
%e A338259 n=3: 1 36  306   996   1446    984    303     42     2
%e A338259 n=4: 1 48  576  2800   6525   7848   4957   1644   274    22    1
%e A338259 n=5: 1 60  930  6020  19365  33600  32487  17694  5336   858   71   2
%e A338259 n=6: 1 72 1368 11064  45435 103200 134806 102912 45567 11358 1510  86  1
%e A338259    ...
%e A338259 Row n=4 corresponds to the polynomial 1 + 48*(1+x) + 576*(1+x)^2 + 2800*(1+x)^3 + 6525*(1+x)^4 + 7848*(1+x)^5 + 4957*(1+x)^6 + 1644*(1+x)^7 + 274*(1+x)^8 + 22*(1+x)^9 + (1+x)^10.
%p A338259 (n,k) -> binomial(n,k)*binomial(12,k)+18*binomial(n+1,k)*binomial(10,k-2)+84*binomial(n+2,k)*binomial(8,k-4)+126*binomial(n+3,k)*binomial(6,k-6)+57*binomial(n+4,k)*binomial(4,k-8)+4*binomial(n+5,k)*binomial(2,k-10) +add(4*binomial(n+1+h,k)*binomial(9,k-3)+24*binomial(n+2+h,k)*binomial(7,k-5)+36*binomial(n+3+h,k)*binomial(5,k-7)+14*binomial(n+4+h,k)*binomial(3,k-9),h = 0 .. 1) +add(add(binomial(2,s)*binomial(2,h)*binomial(n+2+s+h,k)*binomial(6-2*s,k-6-2*s),s = 0 .. 2),h = 0 .. 2)
%Y A338259 Column k=0 is A000012.
%Y A338259 Column k=1 is A008594.
%Y A338259 Row n=3 is identical to row n=4 of A338217 owing to symmetry of hexagonal graphene flakes.
%Y A338259 Row sums give A107915.
%Y A338259 Row sums give column k=0 of A338244.
%K A338259 nonn,tabf
%O A338259 1,2
%A A338259 _Henryk A. Witek_, Oct 19 2020