Henryk A. Witek - cp's OEIS frontend

User: Henryk A. Witek

Henryk A. Witek has authored 4 sequences.

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).

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

Offset: 1

Henryk A. Witek, Oct 19 2020

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).

			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.

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 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).

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.

(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)

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).

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).

Original entry on oeis.org

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

Henryk A. Witek, Oct 18 2020

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).

			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.

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)).

Column k=0 is A107915.

(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).

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).

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).

Original entry on oeis.org

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

Henryk A. Witek, Oct 17 2020

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).

			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.

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).

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.

(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)

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).

A338158 Triangle read by rows: T(n,k) is the coefficient of x^k in the ZZ polynomial of the hexagonal graphene flake O(3,3,n).

Original entry on oeis.org

20, 30, 12, 1, 175, 450, 425, 180, 33, 2, 980, 3308, 4458, 3065, 1140, 225, 22, 1, 4116, 16468, 27293, 24262, 12521, 3796, 653, 58, 2, 14112, 63522, 120848, 126518, 79506, 30681, 7132, 933, 58, 1, 41580, 204180, 429030, 503664, 361690, 163380, 45885, 7588, 648, 20

Offset: 1

Henryk A. Witek, Oct 14 2020

The maximum k for which T(n,k) is nonzero, denoted as 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 Clar covers of order k in the hexagonal graphene flake O(3,3,n).
The Kekulé number of O(3,3,n) is given by T(n, 0).
ZZ polynomials of hexagonal graphene flakes O(3,3,n) with n=1..10 are listed in Eq.(36) of Chou, Li and Witek.
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 also computed using ZZDecomposer (see links below), a graphical program to compute ZZ polynomials of general benzenoids.

			Triangle begins:
       k=0    k=1    k=2    k=3    k=4    k=5   k=6  k=7 k=8 k=9
n=1:    20     30     12      1
n=2:   175    450    425    180     33      2
n=3:   980   3308   4458   3065   1140    225    22    1
n=4:  4116  16468  27293  24262  12521   3796   653   58   2
n=5: 14112  63522 120848 126518  79506  30681  7132  933  58   1
n=6: 41580 204180 429030 503664 361690 163380 45885 7588 648  20
   ...
Row n=4 corresponds to the polynomial 4116 + 16468*x + 27293*x^2 + 24262*x^3 + 12521*x^4 + 3796*x^5 + 653*x^6 + 58*x^7 + 2*x^8.

C.-P. Chou, ZZDecomposer.
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 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)).

Column k=0 is A047819.

Other representation of ZZ polynomials of O(3,3,n) is given by A338217.

(n,k)->add(binomial(i+k,k)*(binomial(9,i+k)*binomial(n,i+k)+(10*binomial(7,i+k-2)-binomial(6,i+k-2))*binomial(n+1,i+k)+(20*binomial(5,i+k-4)+binomial(3,i+k-3)-binomial(3,i+k-5))*binomial(n+2,i+k)+(10*binomial(3,i+k-6)+binomial(2,i+k-5)+binomial(3,i+k-5))*binomial(n+3,i+k)+binomial(2,i+k-7)*binomial(n+4,i+k)),i = 0..9)

T(n,k) = Sum_{l=0..9} C(k+l,k) * (C(9,k+l)*C(n,k+l) + (10*C(7,k+l-2) - C(6,k+l-2)) * C(n+1,k+l) + (20*C(5,k+l-4) + C(3,k+l-3) - C(3,k+l-5)) * C(n+2,k+l) + (10*C(3,k+l-6) + C(2,k+l-5) + C(3,k+l-5)) * C(n+3,k+l) + C(2,k+l-7) * C(n+4,k+l)) where C(n,k) = binomial(n,k). This formula can be obtained by a double sum rotation from Eq.(13) of Witek, Langner, Mos and Chou. Eq.(13) was first discovered heuristically as Eq.(37) of Chou, Li and Witek; a formal proof was given in Eqs.(66-71) on pp. 100-102 of Chou and Witek.

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User: Henryk A. Witek

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).

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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).

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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).

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A338158 Triangle read by rows: T(n,k) is the coefficient of x^k in the ZZ polynomial of the hexagonal graphene flake O(3,3,n).

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