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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). 0
1, 12, 18, 41, 24, 120, 200, 120, 24, 11, 36, 306, 996, 1446, 984, 303, 42, 21, 48, 576, 2800, 6525, 7848, 4957, 1644, 274, 22, 11, 60, 930, 6020, 19365, 33600, 32487, 17694, 5336, 858, 71, 21, 72, 1368, 11064, 45435, 103200, 134806, 102912, 45567, 11358, 1510, 86, 1 (list; graph; refs; listen; history; text; internal format)
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(3,4,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.
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).
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).
LINKS
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 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.
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(3,4,n)).
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(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).
EXAMPLE
Triangle begins:
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
n=1: 1 12 18 4
n=2: 1 24 120 200 120 24 1
n=3: 1 36 306 996 1446 984 303 42 2
n=4: 1 48 576 2800 6525 7848 4957 1644 274 22 1
n=5: 1 60 930 6020 19365 33600 32487 17694 5336 858 71 2
n=6: 1 72 1368 11064 45435 103200 134806 102912 45567 11358 1510 86 1
...
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.
MAPLE
(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)
CROSSREFS
Column k=0 is A000012.
Column k=1 is A008594.
Row n=3 is identical to row n=4 of A338217 owing to symmetry of hexagonal graphene flakes.
Row sums give A107915.
Row sums give column k=0 of A338244.
Sequence in context: A349180 A197464 A124205 * A133403 A152615 A258088
KEYWORD
nonn,tabf
AUTHOR
Henryk A. Witek, Oct 19 2020
STATUS
approved

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Last modified April 12 16:20 EDT 2024. Contains 371635 sequences. (Running on oeis4.)