A002057 -id:A002057 - cp's OEIS frontend

A000063 Symmetrical dissections of an n-gon.

1, 1, 2, 4, 5, 14, 14, 39, 42, 132, 132, 424, 429, 1428, 1430, 4848, 4862, 16796, 16796, 58739, 58786, 208012, 208012, 742768, 742900, 2674426, 2674440, 9694416, 9694845, 35357670, 35357670, 129643318, 129644790, 477638700, 477638700, 1767258328, 1767263190, 6564120288

Offset: 5

N. J. A. Sloane

nonn

This sequence, S_n in Guy's 1958 paper, counts triangulations of a regular n-gon into n-2 triangles such that the only symmetries of the triangulation are the identity and a single reflection ("symmetry of a kite"). Triangulations related by a symmetry of the underlying n-gon do not count as distinct. - Joseph Myers, Jun 21 2012

A000108 is a subsequence, see formula. - Ralf Stephan, Aug 19 2004 (edited, Joerg Arndt, Aug 31 2014)

R. K. Guy, Dissecting a polygon into triangles, Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joseph Myers, Table of n, a(n) for n = 5..1000
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]

c[n_Integer] := CatalanNumber[n]; c[] = 0; a[n] := c[Floor[n/2]-1] - c[n/4-1] - c[n/6-1]; Array[a, 40, 5] (* Jean-François Alcover, Feb 03 2016, after Joseph Myers *)

C(n)=if(type(n)==type(1), binomial(2*n,n)/(n+1), 0);
a(n)=C(floor(n/2)-1) - C(n/4-1) - C(n/6-1);
vector(66,n, a(n+4))
\\ Joerg Arndt, Aug 31 2014

a(2n+3) = A000108(n), n>0. - M. F. Hasler, Mar 25 2012

a(n) = Catalan(floor(n/2) - 1) - Catalan(n/4 - 1) - Catalan (n/6 - 1), where Catalan(x) = 0 for noninteger x (from Guy's 1958 paper). - Joseph Myers, Jun 21 2012

Extended by Joseph Myers, Jun 21 2012

A000131 Number of asymmetrical dissections of n-gon.

Original entry on oeis.org

2, 5, 21, 61, 214, 669, 2240, 7330, 24695, 83257, 284928, 981079, 3410990, 11937328, 42075242, 149171958, 531866972, 1905842605, 6861162880, 24805692978, 90035940227, 327987890608, 1198853954688, 4395797189206, 16165195705544, 59609156824273, 220373268471398, 816677398144221

Offset: 7

N. J. A. Sloane

nonn

This sequence, U_n in Guy's 1958 paper, counts triangulations of a regular n-gon into n-2 triangles with no nonidentity symmetries. Triangulations related by a symmetry of the underlying n-gon do not count as distinct. - Joseph Myers, Jun 21 2012

R. K. Guy, Dissecting a polygon into triangles, Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joseph Myers, Table of n, a(n) for n = 7..1000
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]

Cf. A000063.

catalan[n_] := Block[{c = Binomial[2 n, n]/(n + 1)}, If[IntegerQ[c], c, 0]]; f[n_] := (catalan[n - 2] - (n/2) catalan[n/2 - 1] - n*catalan[Floor[n/2] - 1] - (n/3)*catalan[n/3 - 1] + n*catalan[n/4 - 1] + n*catalan[n/6 - 1])/(2 n); Array[f, 28, 7] (* Robert G. Wilson v, Jun 23 2014 *)

C(n)=if(denominator(n)==1,binomial(2*n,n)/(n+1),0)
a(n)=(C(n-2)/n-C(n/2-1)/2-C(n\2-1)-C(n/3-1)/3+C(n/4-1)+C(n/6-1))/2 \\ Charles R Greathouse IV, Apr 05 2013

a(n) = (Catalan(n-2) - (n/2)*Catalan(n/2 - 1) - n*Catalan(floor(n/2) - 1) - (n/3)*Catalan(n/3 - 1) + n*Catalan(n/4 - 1) + n*Catalan(n/6 - 1))/(2*n), where Catalan(x) = 0 for noninteger x (derived from Guy's 1958 paper). - Joseph Myers, Jun 21 2012

Extended by Joseph Myers, Jun 21 2012

A000912 Expansion of (sqrt(1-4x^2) - sqrt(1-4x))/(2x).

Original entry on oeis.org

1, 0, 2, 4, 14, 40, 132, 424, 1430, 4848, 16796, 58744, 208012, 742768, 2674440, 9694416, 35357670, 129643360, 477638700, 1767258328, 6564120420, 24466250224, 91482563640, 343059554864, 1289904147324, 4861946193440, 18367353072152

Offset: 0

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

nonn

Number of bond-rooted polyenoids with 2n-1 edges.

Partial sums are A129366.

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751

T. D. Noe, Table of n, a(n) for n = 0..200
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

c:=n->binomial(2*n,n)/(n+1):a:=proc(n) if n mod 2 = 1 then c(n+1) else c(n+1)-c(n/2) fi end: seq(a(n),n=0..28); # Emeric Deutsch, Dec 19 2004

nn = 200; CoefficientList[Series[(Sqrt[1 - 4 x^2] - Sqrt[1 - 4 x])/(2 x), {x, 0, nn}], x] (* T. D. Noe, Jun 20 2012 *)
Table[If[EvenQ[n],CatalanNumber[n],CatalanNumber[n]-CatalanNumber[(n-1)/ 2]],{n,0,30}] (* Harvey P. Dale, Oct 30 2013 *)

a(n) = C(n) if n is even and a(n) = C(n) -C((n-1)/2) if n is odd, where C(n) = binomial(2n, n)/(n+1) are the Catalan numbers (A000108). a(n) = 2*A000150(n) for n > 0. - Emeric Deutsch, Dec 19 2004
G.f.: c(x) - x*c(x^2), where c(x) = g.f. for A000108; a(n) = C(n) - C((n-1)/2)(1-(-1)^n)/2, C(n) = A000108(n). - Paul Barry, Apr 11 2007
D-finite with recurrence n*(n+1)*a(n) - 6*n*(n-1)*a(n-1) + 4*(2*n^2-10*n+9)*a(n-2) + 8*(n^2+n-9)*a(n-3) - 48*(n-3)*(n-4)*a(n-4) + 32*(2*n-9)*(n-5)*a(n-5) = 0. - R. J. Mathar, Nov 24 2012

More terms from Emeric Deutsch, Dec 19 2004

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 4, 9, 14, 0, 1, 5, 14, 28, 42, 0, 1, 6, 20, 48, 90, 132, 0, 1, 7, 27, 75, 165, 297, 429, 0, 1, 8, 35, 110, 275, 572, 1001, 1430, 0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 0, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796

Offset: 0

Peter Luschny, Jun 25 2022

The Fuss-Catalan triangle of order m is a regular, (0, 0)-based table recursively defined as follows: Set row(0) = [1] and row(1) = [0, 1]. Now assume row(n-1) already constructed and duplicate the last element of row(n-1). Next apply the cumulative sum m times to this list to get row(n). Here m = 1. (See the Python program for a reference implementation.)

This definition also includes the classical Fuss-Catalan numbers, since T(n, n) = A000108(n), or row 2 in A355262. For m = 2 see A355172 and for m = 3 A355174. More generally, for n >= 1 all Fuss-Catalan sequences (A355262(n, k), k >= 0) are the main diagonals of the Fuss-Catalan triangles of order n - 1.

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1, 2]
  [3] [0, 1, 3,  5]
  [4] [0, 1, 4,  9,  14]
  [5] [0, 1, 5, 14,  28,  42]
  [6] [0, 1, 6, 20,  48,  90,  132]
  [7] [0, 1, 7, 27,  75, 165,  297, 429]
  [8] [0, 1, 8, 35, 110, 275,  572, 1001, 1430]
  [9] [0, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1, 2,  5,  14,   42,  132,   429,  1430,   4862,   16796, ...  A000108
  [1] 0, 1, 3,  9,  28,   90,  297,  1001,  3432,  11934,   41990, ...  A000245
  [2] 0, 1, 4, 14,  48,  165,  572,  2002,  7072,  25194,   90440, ...  A099376
  [3] 0, 1, 5, 20,  75,  275, 1001,  3640, 13260,  48450,  177650, ...  A115144
  [4] 0, 1, 6, 27, 110,  429, 1638,  6188, 23256,  87210,  326876, ...  A115145
  [5] 0, 1, 7, 35, 154,  637, 2548,  9996, 38760, 149226,  572033, ...  A000588
  [6] 0, 1, 8, 44, 208,  910, 3808, 15504, 62016, 245157,  961400, ...  A115147
  [7] 0, 1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, ...  A115148

A000108 (main diagonal), A000245 (subdiagonal), A002057 (diagonal 2), A000344 (diagonal 3), A000027 (column 2), A000096 (column 3), A071724 (row sums), A000958 (alternating row sums), A262394 (main diagonal of array).
Variants: A009766 (main variant), A030237, A130020.
Cf. A123110 (triangle of order 0), A355172 (triangle of order 2), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).
Cf. A115144, A115145, A000588, A115147, A115148.

from functools import cache
from itertools import accumulate
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
    return list(accumulate(row))
for n in range(11): print(Trow(n))

The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
FCT(n, k, m) = (m*(n - k) + m + 1)*(m*n + k - 1)!/((m*n + 1)!*(k - 1)!) for k > 0 and FCT(n, 0, m) = 0^n. The case considered here is T(n, k) = FCT(n, k, 1).
T(n, k) = (n - k + 2)*(n + k - 1)!/((n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
The g.f. of row n of the FC-triangle of order m is 0^n + (x - (m + 1)*x^2) / (1 - x)^(m*n + 2), thus:
T(n, k) = [x^k] (0^n + (x - 2*x^2)/(1 - x)^(n + 2)).

A000913 Number of bond-rooted polyenoids with n edges.

Original entry on oeis.org

0, 1, 2, 12, 38, 143, 490, 1768, 6268, 22610, 81620, 297160, 1086172, 3991995, 14731290, 54587280, 202992808, 757398510, 2834493948, 10637507400, 40023577524, 150946230006, 570534370692, 2160865067312, 8199710635816

Offset: 1

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

nonn

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin, and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin, and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

Cf. A000108 (Catalan).

c:=proc(n) if floor(n)=n then binomial(2*n,n)/(n+1) else 0 fi end:a:=n->(1/4)*c(n+2)-(1/2)*c(n+1)-(3/4)*c((n+1)/2)+(1/2)*c((n-1)/4):seq(a(n),n=1..27); # Emeric Deutsch, Dec 19 2004

c[n_] := If[Floor[n] == n, Binomial[2 n, n]/(n + 1), 0]; a[n_] := (1/4)*c[n + 2] - (1/2)*c[n + 1] - (3/4)*c[(n + 1)/2] + (1/2)*c[(n - 1)/4]; Table[a[n], {n, 1, 25}] (* James C. McMahon, Dec 09 2023 after MAPLE by Emeric Deutsch *)

c(n) = if ((n<0) || (denominator(n) > 1), 0, binomial(2*n,n)/(n+1));
a(n) = (1/4)*c(n+2) - (1/2)*c(n+1) - (3/4)*c((n+1)/2) + (1/2)*c((n-1)/4); \\ Michel Marcus, Aug 26 2019

From Emeric Deutsch, Dec 19 2004: (Start)
a(n) = (1/4)*c(n+2) - (1/2)*c(n+1) - (3/4)*c((n+1)/2) + (1/2)*c((n-1)/4), where c(n) = binomial(2n, n)/(n+1) are the Catalan numbers for n a nonnegative integer and 0 otherwise.
G.f.: (-4x + 8x^2 - sqrt(1-4x) + 2x*sqrt(1-4x) + 3*sqrt(1-4x^2) - 2*sqrt(1-4x^4))/(8x^3). (End)

More terms from Emeric Deutsch, Dec 19 2004

A000936 Number of free planar polyenoids with n nodes and symmetry point group C_{2v}.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 4, 12, 10, 29, 27, 88, 76, 247, 217, 722, 638, 2134, 1901, 6413

Offset: 1

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

a(16)-a(20) and title improved by Sean A. Irvine, Oct 14 2015

A000941 Number of free planar polyenoids with n nodes and symmetry point group C_s.

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 21, 58, 194, 570, 1790, 5434, 16924, 52362, 163784, 512670, 1614406, 5096314, 16150180

Offset: 1

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

Added a(16)-a(19) and title improved by Sean A. Irvine, Oct 13 2015

A000942 Number of free planar polyenoids with n nodes.

Original entry on oeis.org

1, 1, 1, 3, 4, 12, 26, 77, 204, 624, 1817, 5585, 17007, 52803, 164001, 514009, 1615044, 5100324, 16152134, 51324864

Offset: 1

E. K. Lloyd (E.K.Lloyd(AT)soton.ac.uk)

S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]

Cf. A197459 (where polyedges may have cycles).

More terms from Sean A. Irvine, Oct 12 2015

A050163 T(n, k) = S(2n+2, n+2, k+2) for 0<=k<=n and n >= 0, array S as in A050157.

Original entry on oeis.org

1, 3, 4, 9, 14, 15, 28, 48, 55, 56, 90, 165, 200, 209, 210, 297, 572, 726, 780, 791, 792, 1001, 2002, 2639, 2912, 2989, 3002, 3003, 3432, 7072, 9620, 10880, 11320, 11424, 11439, 11440, 11934, 25194, 35190, 40698, 42942, 43605

Offset: 0

Clark Kimberling

			Triangle starts:
                                1
                              3, 4
                           9, 14, 15
                         28, 48, 55, 56
                     90, 165, 200, 209, 210
                  297, 572, 726, 780, 791, 792
            1001, 2002, 2639, 2912, 2989, 3002, 3003

T(n, 0) = A000245(n+1).
T(n, 1) = A002057(n).
T(n, n) = A001791(n+1).
Row sums are A000531(n+1).
Cf. A050155, A050157.

A050163 := (n, k) -> binomial(2*n+2, n) - binomial(2*n+2, n+k+3):
seq(seq(A050163(n,k), k=0..n), n=0..8); # Peter Luschny, Dec 21 2017

T(n, k) = Sum_{0<=j<=k} t(n, j), array t as in A050155.

T(n, k) = binomial(2*n+2, n) - binomial(2*n+2, n+k+3). - Peter Luschny, Dec 21 2017

A118920 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross the x-axis k times (n>=1, k>=0).

Original entry on oeis.org

2, 4, 2, 10, 8, 2, 28, 28, 12, 2, 84, 96, 54, 16, 2, 264, 330, 220, 88, 20, 2, 858, 1144, 858, 416, 130, 24, 2, 2860, 4004, 3276, 1820, 700, 180, 28, 2, 9724, 14144, 12376, 7616, 3400, 1088, 238, 32, 2, 33592, 50388, 46512, 31008, 15504, 5814, 1596, 304, 36, 2

Offset: 1

Emeric Deutsch, May 06 2006

A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1).
Row sums are the central binomial coefficients (A000984). T(n,0)=2*A000108(n) (the Catalan numbers doubled). T(n,1)=2*A002057(n-2). Sum(k*T(n,k),k>=0)=2*A008549(n-1). For crossings of the x-axis in one direction, see A118919.
This triangle is related to paired pattern P_3 and P_4 defined in the Pan & Remmel link. - Ran Pan, Feb 01 2016

			T(3,1)=8 because we have ud|dudu,ud|dduu,udud|du,uudd|du,du|udud,du|uudd, dudu|ud and dduu|ud (the crossings of the x-axis are shown by |).
Triangle starts:
   2;
   4, 2;
  10, 8, 2;
  28,28,12, 2;
  84,96,54,16,2;

Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Cf. A000984, A000108, A002057, A008549, A039598, A118919.

T:=(n,k)->2*(k+1)*binomial(2*n,n-k-1)/n: for n from 1 to 11 do seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form

Table[2 (k + 1) Binomial[2 n, n - k - 1]/n, {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Feb 01 2016 *)

T(n,k) = 2*(k+1)*binomial(2*n,n-k-1)/n \\ Charles R Greathouse IV, Feb 01 2016

# Algorithm of L. Seidel (1877)
# Prints the first n rows of the triangle.
def A118920_triangle(n) :
    D = [0]*(n+2); D[1] = 2
    b = True; h = 1
    for i in range(2*n) :
        if b :
            for k in range(h, 0, -1) : D[k] += D[k-1]
            h += 1
        else :
            for k in range(1, h, 1) : D[k] += D[k+1]
        b = not b
        if b : print([D[z] for z in (1..h-1)])
A118920_triangle(10)  # Peter Luschny, Oct 19 2012

T(n,k) = 2*(k+1)*binomial(2*n,n-k-1)/n.
G.f.: G(t,z)=2*z*C^2/(1-t*z*C^2), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function.
More generally, the trivariate g.f. G=G(x,y,z), where x (y) marks number of downward (upward) crossings of the x-axis, is given by G = z*C^2*(2+(x+y)*z*C^2)/(1-x*y*z^2*C^4).
a(n) = 2 * A039598(n-1). - Georg Fischer, Oct 27 2021

cp's OEIS Frontend

A000063 Symmetrical dissections of an n-gon.

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A000131 Number of asymmetrical dissections of n-gon.

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A000912 Expansion of (sqrt(1-4x^2) - sqrt(1-4x))/(2x).

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A355173 The Fuss-Catalan triangle of order 1, read by rows. Related to binary trees.

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A000913 Number of bond-rooted polyenoids with n edges.

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A000936 Number of free planar polyenoids with n nodes and symmetry point group C_{2v}.

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A000941 Number of free planar polyenoids with n nodes and symmetry point group C_s.

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A000942 Number of free planar polyenoids with n nodes.

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