A047819 -id:A047819 - cp's OEIS frontend

A296419 Triangle T(i,j) read by rows: Number of plane bipolar orientations with i+1 vertices and j+1 faces.

1, 1, 4, 1, 10, 50, 1, 20, 175, 980, 1, 35, 490, 4116, 24696, 1, 56, 1176, 14112, 116424, 731808, 1, 84, 2520, 41580, 457380, 3737448, 24293412, 1, 120, 4950, 108900, 1557270, 16195608, 131589315, 877262100, 1, 165, 9075, 259545, 4723719, 61408347, 614083470, 4971151900, 33803832920

Offset: 1

R. J. Mathar, Feb 25 2018

			The triangle starts in row 1 as
  1;
  1,   4;
  1,  10,   50;
  1,  20,  175,    980;
  1,  35,  490,   4116,   24696;
  1,  56, 1176,  14112,  116424,   731808;
  1,  84, 2520,  41580,  457380,  3737448,  24293412;
  1, 120, 4950, 108900, 1557270, 16195608, 131589315, 877262100;

E. Fusy, D. Poulalhon, and G. Schaeffer, Bijective counting of plane bipolar orientations, El. Notes Discr. Math. 29 (2007) 283-287.

Cf. rows/columns: A006542, A047819, A107915, A140901, A140903, A140907.

A296419 := proc(i,j)
    2*(i+j-2)!*(i+j-1)!*(i+j)!/(i-1)!/i!/(i+1)!/(j-1)!/j!/(j+1)! ;
end proc:
seq(seq(A296419(i,j),j=1..i),i=1..10) ;

T(i,j) = T(j,i) = 2*(i+j-2)!*(i+j-1)!*(i+j)!/((i-1)!*i!*(i+1)!*(j-1)!*j!*(j+1)!).

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).

A057658 a(n) = n(n+1)^2(n+2)^3(n+3)^2(n+4).

Original entry on oeis.org

0, 8640, 172800, 1512000, 8467200, 35562240, 121927680, 359251200, 940896000, 2242468800, 4947022080, 10231341120, 20033395200, 37425024000, 67118284800, 116138603520, 194702952960, 317346724800, 504348768000, 783510235200

Offset: 0

N. J. A. Sloane, Oct 16 2000

nonn

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45, 10,-1).

Cf. A047819.

[n*(n+1)^2*(n+2)^3*(n+3)^2*(n+4): n in [0..25]]; // Vincenzo Librandi, Jun 07 2019

seq(n*(n+1)^2*(n+2)^3*(n+3)^2*(n+4), n=0..30); # Robert Israel, Jun 06 2019

Table[n (n+1)^2 (n+2)^3 (n+3)^2 (n+4), {n, 0, 40}] (* Vincenzo Librandi, Jun 07 2019 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,8640,172800,1512000,8467200,35562240,121927680,359251200,940896000,2242468800},30] (* Harvey P. Dale, Sep 24 2021 *)

From Robert Israel, Jun 06 2019: (Start)
G.f.: 8640*x*(x^4 + 10*x^3 + 20*x^2 + 10*x + 1)/(x - 1)^10.
(n + 3)*(n + 2)*a(n - 2) - 2*(n^2 + 2*n + 12)*a(n - 1) + n*(n - 1)*a(n) = 0. (End)
a(n) = 8640*A047819(n) for n > 0. - Michel Marcus, Jun 07 2019

A133815 Square array of Hankel transforms of binomial(n+k,floor((n+k)/2)), read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, -1, 2, 1, 1, -1, 3, 3, 1, 1, 1, 4, -6, 6, 1, 1, 1, 5, -10, 20, 10, 1, 1, -1, 6, 15, 50, -50, 20, 1, 1, -1, 7, 21, 105, -175, 175, 35, 1, 1, 1, 8, -28, 196, 490, 980, -490, 70, 1, 1, 1, 9, -36, 336, 1176, 4116, -4116, 1764, 126, 1

Offset: 0

Paul Barry, Sep 24 2007

T(n+1,k) is the Hankel transform of binomial(n+k, floor((n+k)/2)).

Even-indexed columns count tilings of hexagons: A002415 (<2,n,2>), A047819 (<3,n,3>), A047835 (<4,n,4>), etc.

			Array begins
  1,    1,    1,    1,    1,    1, ...
  1,    1,    2,    3,    6,   10, ...
  1,   -1,    3,   -6,   20,  -50, ...
  1,   -1,    4,  -10,   50, -175, ...
  1,    1,    5,   15,  105,  490, ...
  1,    1,    6,   21,  196, 1176, ...
As a number triangle, T(n-k,k) gives
  1;
  1,   1;
  1,   1,   1;
  1,  -1,   2,   1;
  1,  -1,   3,   3,   1;
  1,   1,   4,  -6,   6,   1;
  1,   1,   5, -10,  20,  10,   1;
  1,  -1,   6,  15,  50, -50,  20,   1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A002415, A047819, A047835, A103905, A120247.

F:= Floor;
function t(n,k)
  if k eq 0 then return 1;
  elif k eq 1 then return (-1)^F(n/2);
  elif (k mod 2) eq 0 then return (&*[ Binomial(n+F(k/2)+j, F(k/2))/Binomial(F(k/2)+j, F(k/2)) : j in [0..F((k-2)/2)] ]);
  else return (-1)^F(n/2)*(&*[ Binomial(n+F((k+1)/2)+j, F((k+1)/2))/Binomial(F((k+1)/2)+j, F((k+1)/2)) : j in [0..F((k-3)/2)] ]);
  end if;
end function;
// [[t(n,k): k in [0..10]]: n in [0..10]];
A133815:= func< n,k | t(n-k, k) >;
[A133815(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 16 2023

T[ n_, m_] := With[{k = Quotient[m + 1, 2]}, (-1)^(Quotient[n, 2] m) Product[ Binomial[n + k + j, k] / Binomial[k + j, k], {j, 0, k - 1 - Mod[m, 2]}]];
(* Michael Somos, Apr 03 2021 *)

alias(C, binomial);
T(n,k) = if (k % 2 == 0, prod(j=0, (k-2)/2, C(n+k/2+j,k/2)/C(k/2+j,k/2)), (cos(Pi*n/2)+sin(Pi*n/2))*prod(j=0, (k-3)/2, C(n+(k+1)/2+j,(k+1)/2)/C((k+1)/2+j,(k+1)/2)));
tabl(nn) = matrix(nn, nn, n, k, round(T(n-1, k-1))); \\ Michel Marcus, Dec 10 2016

T(n, m) = my(k = (m+1)\2); (-1)^(n\2*m) * prod(j=0, k-1-m%2, binomial(n+k+j, k) / binomial(k+j, k)); /* Michael Somos, Apr 03 2021 */

def f(k): return (k+1)//2
def t(n, k): return (-1)^(k*(n//2))*product(binomial(n+f(k) +j, f(k))/binomial(f(k) +j, f(k)) for j in range(f(k-1)))
def A133815(n,k): return t(n-k, k)
flatten([[A133815(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 16 2023

T(n,k) = if(k mod 2 = 0, Product_{j=0..(k-2)/2} C(n+k/2+j,k/2) / C(k/2+j,k/2), (cos(Pi*n/2) + sin(Pi*n/2))*Product_{j=0..(k-3)/2} C(n+(k+1)/2+j,(k+1)/2)/C((k+1)/2+j,(k+1)/2)).

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.

cp's OEIS Frontend

A296419 Triangle T(i,j) read by rows: Number of plane bipolar orientations with i+1 vertices and j+1 faces.

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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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A057658 a(n) = n*(n+1)^2*(n+2)^3*(n+3)^2*(n+4).

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A133815 Square array of Hankel transforms of binomial(n+k,floor((n+k)/2)), read by antidiagonals.

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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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Maple

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A057658 a(n) = n(n+1)^2(n+2)^3(n+3)^2(n+4).