A338244 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).

35, 60, 30, 4, 490, 1470, 1695, 940, 255, 30, 1, 4116, 16468, 27293, 24262, 12521, 3796, 653, 58, 2, 24696, 118590, 243994, 281372, 199822, 90482, 26195, 4748, 517, 32, 10, 116424, 635362, 1513660, 2068248, 1791158, 1025836, 393659, 100450, 16583, 1678, 930, 21

Offset: 1

Author

Henryk A. Witek, Oct 18 2020

Keywords

• nonn
• tabf

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 Clar covers of order k in the hexagonal graphene flake O(3,4,n).

The Kekulé number of O(3,4,n) is given by T(n, 0).

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).

Examples

			Triangle begins:
        k=0    k=1     k=2     k=3     k=4     k=5    k=6    k=7   k=8  k=9 k=10
n=1:     35     60      30       4
n=2:    490   1470    1695     940     255      30      1
n=3:   4116  16468   27293   24262   12521    3796    653     58     2
n=4:  24696 118590  243994  281372  199822   90482  26195   4748   517   32  1
n=5: 116424 635362 1513660 2068248 1791158 1025836 393659 100450 16583 1678 93 2
   ...
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.
		

Links

• 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 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 I, Discrete Appl. Math. 69 (1996), 147-167 (ZZ polynomial is defined by Eq.(2.1) and working formula is given by Eq.(2.2)).

Crossrefs

Column k=0 is A107915.

Programs

Maple

(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).

Formula

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).