A107915 -id:A107915 - cp's OEIS frontend

A047819 a(n) = Product_{i=1..n} ((i+3)(i+4)(i+5))/(i(i+1)(i+2)).

1, 20, 175, 980, 4116, 14112, 41580, 108900, 259545, 572572, 1184183, 2318680, 4331600, 7768320, 13441968, 22535064, 36729945, 58373700, 90684055, 138003404, 206108980, 302588000, 437287500, 622849500, 875343105, 1215006156, 1667110095

Offset: 0

N. J. A. Sloane

Number of tilings of a <3,n,3> hexagon.
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 18 2005
Partial sums of A107891. - Peter Bala, Sep 21 2007
Determinant of the 3 X 3 matrix with rows [C(n+3,3) C(n+3,4) C(n+3,5)], [C(n+4,3) C(n+4,4) C(n+4,5)], and [C(n+5,3) C(n+5,4) C(n+5,5)]. - J. M. Bergot, Sep 10 2013

			G.f. = 1 + 20*x + 175*x^2 + 980*x^3 + 4116*x^4 + 14112*x^5 + 41580*x^6 + ...

O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 232, # 2 and p. 105, eq.(ii), K(0a(2,5,n))).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Paolo Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
O. D. Anderson, Find the next sequence, J. Rec. Math., 8 (No. 4, 1975-1976), 241. [Annotated scanned copy]
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.2(ii), case a=2].
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 25.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Third row of array A103905.

Cf. A002415, A107891, A107915, A133708.

a:=n->(n+1)*(n+2)^2*(n+3)^3*(n+4)^2*(n+5)/8640: seq(a(n),n=0..30); # Emeric Deutsch, Jun 18 2005

a[n_] :=(n + 1)*(n + 2)^2*(n + 3)^3*(n + 4)^2*(n + 5)/8640;
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 19 2018, after Emeric Deutsch *)

{a(n) = if( n<0, 0, binomial(n+5, 5) * binomial(n+4, 3) * (n+3) / 12)}; /* Michael Somos, Nov 14 2002 */

{a(n) = my(s=sign(n+3)); n=abs(n+3)-3; -s/8 * polcoeff( charpoly( matrix(n+3, n+3, i, j, (i-j)^2)), n)}; /* Michael Somos, Nov 14 2002 */

G.f.: (1 + 10*x + 20*x^2 + 10*x^3 + x^4) / (1 - x)^10. - Michael Somos, Nov 14 2002
a(n) = C(n+3,n+2)*C(n+4,n+1)*C(n+5,n)/12. - Zerinvary Lajos, May 29 2007
a(n-3) = (1/24)*Sum_{1 <= x_1, x_2, x_3 <= n} (det V(x_1,x_2,x_3))^2 = (1/24)*Sum_ {1 <= i,j,k <= n} ((i-j)(i-k)(j-k))^2, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. - Peter Bala, Sep 21 2007
a(n) = -a(-6-n) for all n in Z. - Michael Somos, Dec 26 2016
From Amiram Eldar, May 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 5195/2 - 2160*zeta(3).
Sum_{n>=0} (-1)^n/a(n) = 17205/2 - 9600*log(2) - 1620*zeta(3). (End)

A133708 First differences of A047835.

Original entry on oeis.org

0, 0, 1, 69, 1694, 22932, 208152, 1413720, 7697052, 35194302, 139687119, 493127635, 1577331756, 4637757488, 12679063488, 32529562560, 78917794128, 182184724908, 402332471541, 853769650041, 1747606106554, 3462012537060, 6656436729800, 12452933493000

Offset: 1

Peter Bala, Sep 21 2007

nonn

T. D. Noe, Table of n, a(n) for n = 1..1003

Cf. A040977, A047819, A047835, A107915.

Join[{0,0,1},Differences[Table[Product[Times@@((i+Range[4,7])/(i+Range[0,3])),{i,n}],{n,0,30}]]] (* Harvey P. Dale, Aug 08 2015 *)

In terms of Vandermonde determinants, a(n) = 1/864*sum {1 <= x_1,x_2,x_3 <= n} (x_1*x_2*x_3)^2*(det V(x_1,x_2,x_3))^2 = 1/864 *sum {1 <= i,j,k <= n} (i*j*k*(i-j)(i-k)(j-k))^2, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. a(n) = (n-2)*(n-1)^2*n^3*(n+1)^3*(n+2)^2*(n+3)*(n^2+n+3)*(2n+1)/108864000.

Empirical G.f.: x^3*(x+1)*(x^8+52*x^7+658*x^6+2890*x^5+4810*x^4+2890*x^3+658*x^2+52*x+1)/(x-1)^16. [Colin Barker, Jun 06 2012]

A120247 Triangle of Hankel transforms of binomial(n+k, k).

Original entry on oeis.org

1, 1, -1, 1, -3, -1, 1, -6, -10, 1, 1, -10, -50, 35, 1, 1, -15, -175, 490, 126, -1, 1, -21, -490, 4116, 5292, -462, -1, 1, -28, -1176, 24696, 116424, -60984, -1716, 1, 1, -36, -2520, 116424, 1646568, -3737448, -736164, 6435, 1, 1, -45, -4950, 457380, 16818516, -133613766, -131589315, 9202050, 24310, -1

Offset: 0

Paul Barry, Jun 12 2006

Columns include -A000217, -A006542, A107915.
Row k is the Hankel transform of C(n+k, k).
The matrix inverse starts
          1;
          1,       -1;
         -2,        3,        -1;
        -15,       24,       -10,       1;
        434,     -700,       300,     -35,      1;
      47670,   -76950,     33075,   -3920,    126,  -1;
  -19787592, 31943835, -13733720, 1629936, -52920, 462, -1; - R. J. Mathar, Mar 22 2013

			Triangle begins
  1;
  1,  -1;
  1,  -3,    -1;
  1,  -6,   -10,     1;
  1, -10,   -50,    35,      1;
  1, -15,  -175,   490,    126,     -1;
  1, -21,  -490,  4116,   5292,   -462,    -1;
  1, -28, -1176, 24696, 116424, -60984, -1716,   1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Columns include: A000217, A006542, A107915.

Cf. A120248.

p:= func< m,k | k eq 0 select 1 else (&*[Binomial(m+j, k+1): j in [1..k]]) >;
A120247:= func< n,k | (-1)^Floor((k+1)/2)*p(n,k)/p(k,k) >;
[A120247(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 15 2023

A120247 := proc(n,k)
    (cos(Pi*k/2)-sin(Pi*k/2))*mul(binomial(n+j+1,k+1),j=0..k-1)/mul(binomial(k+j+1,k+1),j=0..k-1) ;
    simplify(%) ;
end proc: # R. J. Mathar, Mar 22 2013

p[m_, k_]:= Product[Binomial[m+j, k+1], {j,k}];
T[n_, k_]:= (-1)^Floor[(k+1)/2]*p[n,k]/p[k,k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 15 2023 *)

def p(m,k): return product(binomial(m+j+1,k+1) for j in range(k))
def A120247(n,k): return (-1)^((k+1)//2)*p(n,k)/p(k,k)
flatten([[A120247(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 15 2023

T(n, k) = (cos(pi*k/2) - sin(pi*k/2))*( Product{j=0..k-1} C(n+j+1, k+1)/Product{j=0..k-1} C(k+j+1, k+1) ).

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

Original entry on oeis.org

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)!).

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).

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).

Original entry on oeis.org

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).

cp's OEIS Frontend

A047819 a(n) = Product_{i=1..n} ((i+3)*(i+4)*(i+5))/(i*(i+1)*(i+2)).

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A133708 First differences of A047835.

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A120247 Triangle of Hankel transforms of binomial(n+k, k).

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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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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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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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Maple

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A047819 a(n) = Product_{i=1..n} ((i+3)(i+4)(i+5))/(i(i+1)(i+2)).