A338217 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).

1, 9, 9, 1, 1, 18, 63, 68, 23, 2, 1, 27, 162, 350, 310, 114, 15, 1, 1, 36, 306, 996, 1446, 984, 303, 42, 2, 1, 45, 495, 2155, 4360, 4360, 2141, 505, 49, 1, 1, 54, 729, 3976, 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

Offset

1,2

Comments

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.

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).

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.

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).

Links

Table of n, a(n) for n=1..67.

C.-P. Chou, ZZDecomposer executable.

C.-P. Chou, ZZCalculator source code.

C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH Commun. Math. Comput. Chem. 68 (2012), 3-30.

C.-P. Chou, Y. Li and H. A. Witek, Zhang-Zhang Polynomials of Various Classes of Benzenoid Systems, MATCH Commun. Math. Comput. Chem. 68 (2012), 31-64.

C.-P. Chou and H. A. Witek, ZZDecomposer: A Graphical Toolkit for Analyzing the Zhang-Zhang Polynomials of Benzenoid Structures, MATCH Commun. Math. Comput. Chem. 71 (2014), 741-764.

C.-P. Chou and H. A. Witek, Determination of Zhang-Zhang Polynomials for various Classes of Benzenoid Systems: Non-Heuristic Approach, MATCH Commun. Math. Comput. Chem. 72 (2014), 75-104.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 105 for a graphical definition of O(3,3,n)).

H. A. Witek, J. Langner, G. Mos, and C.-P. Chou, Zhang-Zhang Polynomials of Regular 5-tier Benzeonid Strips, MATCH Commun. Math. Comput. Chem. 78 (2017), 487-504.

H. Zhang and F. Zhang, The Clar covering polynomial of hexagonal systems III, 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).

Formula

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).

Example

Triangle begins:
    k=0 k=1   k=2    k=3    k=4    k=5    k=6    k=7    k=8  k=9
n=1:  1   9     9      1
n=2:  1  18    63     68     23      2
n=3:  1  27   162    350    310    114     15      1
n=4:  1  36   306    996   1446    984    303     42      2
n=5:  1  45   495   2155   4360   4360   2141    505     49    1
n=6:  1  54   729   3976  10325  13650   9233   3124    468   20
n=7:  1  63  1008   6608  20958  34482  29750  13170   2685  175
n=8:  1  72  1332  10200  38220  75264  79002  43284  11190  980
   ...
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.

Maple

(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)

Crossrefs

Column k=0 is A000012.

Column k=1 is A008591.

Column k=2 is 9*A000566.

Row sums give A047819.

Row sums give column k=0 of A338158.

Another representation is given by A338158.

Sequence in context: A197401 A197350 A197364 * A174266 A351722 A334425

Adjacent sequences: A338214 A338215 A338216 * A338218 A338219 A338220

Keyword

nonn,tabf

Author

Henryk A. Witek, Oct 17 2020

Status

approved