A338217 - OEIS

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

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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