A295259 -id:A295259 - cp's OEIS frontend

A295260 Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals up to rotation and reflection (k >= 3).

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 5, 4, 1, 1, 3, 8, 16, 12, 1, 1, 3, 12, 33, 60, 27, 1, 1, 4, 16, 68, 194, 261, 82, 1, 1, 4, 21, 112, 483, 1196, 1243, 228, 1, 1, 5, 27, 183, 1020, 3946, 8196, 6257, 733, 1, 1, 5, 33, 266, 1918, 10222, 34485, 58140, 32721, 2282

Offset: 1

Andrew Howroyd, Nov 18 2017

The polygon prior to dissection will have n*(k-2)+2 sides.
In the Harary, Palmer and Read reference these are the sequences called h.
T(n,k) is the number of unoriented polyominoes containing n k-gonal tiles of the hyperbolic regular tiling with Schläfli symbol {k,oo}. Stereographic projections of several of these tilings on the Poincaré disk can be obtained via the Christensson link. For unoriented polyominoes, chiral pairs are counted as one. T(n,2) could represent polyominoes of the Euclidean regular tiling with Schläfli symbol {2,oo}; T(n,2) = 1. - Robert A. Russell, Jan 21 2024

			Array begins:
  ===================================================
  n\k|   3     4      5       6        7        8
  ---|-----------------------------------------------
   1 |   1     1      1       1        1        1 ...
   2 |   1     1      1       1        1        1 ...
   3 |   1     2      2       3        3        4 ...
   4 |   3     5      8      12       16       21 ...
   5 |   4    16     33      68      112      183 ...
   6 |  12    60    194     483     1020     1918 ...
   7 |  27   261   1196    3946    10222    22908 ...
   8 |  82  1243   8196   34485   109947   290511 ...
   9 | 228  6257  58140  315810  1230840  3844688 ...
  10 | 733 32721 427975 2984570 14218671 52454248 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
Wikipedia, Fuss-Catalan number

Columns k=3..7 are A000207, A005036, A005040, A004127, A005419.
Cf. A033282, A295222, A295259.
Polyominoes: A295224 (oriented), A070914 (rooted).

u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
T[n_, k_] := (u[n, k, 1] + If[OddQ[n], u[(n - 1)/2, k, Quotient[k, 2]], If[OddQ[k], (u[n/2 - 1, k, k - 1] + u[n/2, k, 1])/2, u[n/2, k, 1]]] + (If[EvenQ[n], u[n/2, k, 1]] - u[n, k, 2])/2 + DivisorSum[GCD[n - 1, k], EulerPhi[#]*u[(n - 1)/#, k, k/#] &]/k)/2 /. Null -> 0;
Table[T[n - k + 2, k + 1], {n, 1, 11}, {k, n + 1, 2, -1}] // Flatten (* Jean-François Alcover, Dec 28 2017, after Andrew Howroyd *)

\\ here u is Fuss-Catalan sequence with p = k+1.
u(n,k,r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n,k) = {(u(n,k,1) + if(n%2, u((n-1)/2,k,k\2), if(k%2, (u(n/2-1,k,(k-1)) + u(n/2,k,1))/2, u(n/2,k,1))) + (if(n%2==0, u(n/2,k,1))-u(n,k,2))/2 + sumdiv(gcd(n-1,k), d, eulerphi(d)*u((n-1)/d,k,k/d))/k)/2}
for(n=1, 10, for(k=3, 8, print1(T(n, k), ", ")); print);

T(n,k) ~ A295222(n,k)/(2*n) for fixed k.

A295224 Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals up to rotation (k >= 3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 2, 7, 6, 1, 1, 3, 12, 25, 19, 1, 1, 3, 19, 57, 108, 49, 1, 1, 4, 26, 118, 366, 492, 150, 1, 1, 4, 35, 203, 931, 2340, 2431, 442, 1, 1, 5, 46, 332, 1989, 7756, 16252, 12371, 1424, 1, 1, 5, 57, 494, 3766, 20254, 68685, 115940, 65169, 4522

Offset: 1

Andrew Howroyd, Nov 17 2017

The polygon prior to dissection will have n*(k-2)+2 sides.
In the Harary, Palmer and Read reference these are the sequences called H.
T(n,k) is the number of oriented polyominoes containing n k-gonal tiles of the hyperbolic regular tiling with Schläfli symbol {k,oo}. Stereographic projections of several of these tilings on the Poincaré disk can be obtained via the Christensson link. For oriented polyominoes, chiral pairs are counted as two. T(n,2) could represent polyominoes of the Euclidean regular tiling with Schläfli symbol {2,oo}; T(n,2) = 1. - Robert A. Russell, Jan 21 2024

			Array begins:
  =====================================================
  n\k|    3     4      5       6        7         8
  ---|-------------------------------------------------
   1 |    1     1      1       1        1         1 ...
   2 |    1     1      1       1        1         1 ...
   3 |    1     2      2       3        3         4 ...
   4 |    4     7     12      19       26        35 ...
   5 |    6    25     57     118      203       332 ...
   6 |   19   108    366     931     1989      3766 ...
   7 |   49   492   2340    7756    20254     45448 ...
   8 |  150  2431  16252   68685   219388    580203 ...
   9 |  442 12371 115940  630465  2459730   7684881 ...
  10 | 1424 65169 854981 5966610 28431861 104898024 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
Wikipedia, Fuss-Catalan number

Columns k=3..6 are A001683(n+2), A005034, A005038, A221184(n-1).
Cf. A033282, A070914, A295222, A295259, A295260.
Polyominoes: A295260 (unoriented), A070914 (rooted).

u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
T[n_, k_] := u[n, k, 1] + (If[EvenQ[n], u[n/2, k, 1], 0] - u[n, k, 2])/2 + DivisorSum[GCD[n - 1, k], EulerPhi[#]*u[(n - 1)/#, k, k/#]&]/k;
Table[T[n - k + 1, k], {n, 1, 13}, {k, n, 3, -1}] // Flatten (* Jean-François Alcover, Nov 21 2017, after Andrew Howroyd *)

\\ here u is Fuss-Catalan sequence with p = k+1.
u(n, k, r)={r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n,k) = u(n,k,1) + (if(n%2==0, u(n/2,k,1))-u(n,k,2))/2 + sumdiv(gcd(n-1,k), d, eulerphi(d)*u((n-1)/d,k,k/d))/k;
for(n=1, 10, for(k=3, 8, print1(T(n, k), ", ")); print);

from sympy import binomial, gcd, totient, divisors
def u(n, k, r): return r*binomial((k - 1)*n + r, n)//((k - 1)*n + r)
def T(n, k): return u(n, k, 1) + ((u(n//2, k, 1) if n%2==0 else 0) - u(n, k, 2))//2 + sum([totient(d)*u((n - 1)//d, k, k//d) for d in divisors(gcd(n - 1, k))])//k
for n in range(1, 11): print([T(n, k) for k in range(3, 9)]) # Indranil Ghosh, Dec 13 2017, after PARI code

T(n,k) ~ A295222(n,k)/n for fixed k.

A295222 Array read by antidiagonals: T(n,k) is the number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell up to rotation (k >= 3).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 6, 22, 30, 1, 1, 8, 40, 116, 99, 1, 1, 9, 64, 285, 612, 335, 1, 1, 11, 92, 578, 2126, 3399, 1144, 1, 1, 12, 126, 1015, 5481, 16380, 19228, 3978, 1, 1, 14, 166, 1641, 11781, 54132, 129456, 111041, 14000

Offset: 1

Andrew Howroyd, Nov 17 2017

The polygon prior to dissection will have n*(k-2)+2 sides.

In the Harary, Palmer and Read reference these are the sequences called F.

			Array begins:
  ===========================================================
  n\k|     3      4       5        6         7          8
  ---|-------------------------------------------------------
   1 |     1      1       1        1         1          1 ...
   2 |     1      1       1        1         1          1 ...
   3 |     3      5       6        8         9         11 ...
   4 |    10     22      40       64        92        126 ...
   5 |    30    116     285      578      1015       1641 ...
   6 |    99    612    2126     5481     11781      22386 ...
   7 |   335   3399   16380    54132    141778     317860 ...
   8 |  1144  19228  129456   548340   1753074    4638348 ...
   9 |  3978 111041 1043460  5672645  22137570   69159400 ...
  10 | 14000 650325 8544965 59653210 284291205 1048927880 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
Wikipedia, Fuss-Catalan number

Columns k=3..5 are A003441, A005033, A005037.

Cf. A033282, A070914, A295224, A295259, A295260.

u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
T[n_, k_] := DivisorSum[GCD[n-1, k], EulerPhi[#]*u[(n-1)/#, k, k/#]&]/k;
Table[T[n - k + 3, k], {n, 1, 10}, {k, n + 2, 3, -1}] // Flatten (* Jean-François Alcover, Nov 21 2017, after Andrew Howroyd *)

\\ here u is Fuss-Catalan sequence with p = k+1.
u(n,k,r)={r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n,k)=sumdiv(gcd(n-1,k), d, eulerphi(d)*u((n-1)/d, k, k/d))/k;
for(n=1, 10, for(k=3, 8, print1(T(n, k), ", ")); print);

from sympy import binomial, gcd, totient, divisors
def u(n, k, r): return r*binomial((k - 1)*n + r, n)//((k - 1)*n + r)
def T(n, k): return sum([totient(d)*u((n - 1)//d, k, k//d) for d in divisors(gcd(n - 1, k))])//k
for n in range(1, 11): print([T(n, k) for k in range(3, 9)]) # Indranil Ghosh, Dec 13 2017, after PARI

T(n,k) = Sum_{d|gcd(n-1,k)} phi(d)*u((n-1)/d, k, k/d)/k where u(n,k,r) = r*binomial((k - 1)*n + r, n)/((k - 1)*n + r).

T(n,k) ~ n*A070914(n,k-2)/(n*(k-2)+2) for fixed k.

A003456 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

1, 2, 5, 17, 62, 275, 1272, 6225, 31075, 158376, 816229, 4251412, 22319056, 117998524, 627573216, 3355499036, 18025442261, 97239773408, 526560862829, 2861189112867, 15595669996482, 85252072993968, 467247847612316, 2567091151780343

Offset: 3

N. J. A. Sloane

nonn

Total number of dissections of an n-gon into polygons with reflection and rooted at a cell. - Sean A. Irvine, May 14 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 3..200
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.

Cf. A001004, A003454, A003455, A005035, A295259.

\\ See A003447 for DissectionsModDihedralRooted()
DissectionsModDihedralRooted(apply(i->1, [1..30]))

More terms from Sean A. Irvine, May 14 2015
Name clarified by Andrew Howroyd, Nov 24 2017
a(15) corrected by Andrew Howroyd, Nov 24 2017

A370060 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell, n >= 1, k >= 3.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 4, 2, 1, 1, 4, 4, 12, 5, 1, 1, 3, 6, 9, 18, 5, 1, 1, 5, 6, 26, 22, 55, 14, 1, 1, 4, 8, 21, 45, 52, 88, 14, 1, 1, 6, 8, 45, 51, 204, 140, 273, 42, 1, 1, 5, 10, 38, 84, 190, 380, 340, 455, 42, 1, 1, 7, 10, 69, 92, 500, 506, 1771, 969, 1428, 132

Offset: 1

Andrew Howroyd, Feb 08 2024

The polygon prior to dissection will have n*(k-2)+2 sides.

			Array begins:
=============================================
n\k|  3   4   5    6    7    8    9    10 ...
---+-----------------------------------------
1  |  1   1   1    1    1    1    1     1 ...
2  |  1   1   1    1    1    1    1     1 ...
3  |  1   3   2    4    3    5    4     6 ...
4  |  2   4   4    6    6    8    8    10 ...
5  |  2  12   9   26   21   45   38    69 ...
6  |  5  18  22   45   51   84   92   135 ...
7  |  5  55  52  204  190  500  468   992 ...
8  | 14  88 140  380  506 1008 1240  2100 ...
9  | 14 273 340 1771 1950 6200 6545 15990 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
Wikipedia, Fuss-Catalan number

Columns k=3..6 are A208355(n-1), A124817(n-1), A369472, A370061.

Cf. A070914 (rooted), A295222 (oriented), A295259 (unoriented), A369929, A370062 (achiral unrooted).

\\ here u is Fuss-Catalan sequence with p = k-1.
u(n, k, r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n, k) = {if(k%2, if(n%2, u((n-1)/2, k, (k-1)/2), u(n/2-1, k, (k-1))), if(n%2, u((n-1)/2, k, k/2+1), u(n/2-1, k, k)) )}
for(n=1, 9, for(k=3, 10, print1(T(n, k), ", ")); print);

T(n,k) = 2*A295259(n,k) - A295222(n,k).

T(n,2*k+1) = A370062(n,2*k+1).

A003446 Number of nonequivalent dissections of a polygon into n triangles by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

0, 1, 1, 2, 6, 16, 52, 170, 579, 1996, 7021, 24892, 89214, 321994, 1170282, 4277352, 15715249, 57999700, 214939846, 799478680, 2983699498, 11169391168, 41929537871, 157807451672, 595340479694, 2250901216266, 8527700012092, 32369067177176

Offset: 0

N. J. A. Sloane

Original name: Triangulated (n+2)-gons rooted at one of the triangles.

Also, the total number of atom-rooted polyenoids. - Sean A. Irvine, Oct 05 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, 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.
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]
F. Harary, E. M. Palmer, R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388, Table 1.
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]

Column k=3 of A295259.

c[x_] = (1 - Sqrt[1 - 4*x])/(2*x); d[x_] = 1 + x*c[x^2]; f[x_] = (x/6)*(c[x]^3 + 2*c[x^3] + 3*d[x]*c[x^2]); CoefficientList[ Series[ f[x], {x, 0, 27}], x] (* Jean-François Alcover, Sep 30 2011, after g.f. *)

Let c(x) = (1-sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers (A000108), let d(x) = 1+x*c(x^2). Then g.f. is (x/6)*(c^3+2*subs(x=x^3, c)+3*d*subs(x=x^2, c)).
Recurrence: n*(n+1)*(n+2)*(12*n^10 - 396*n^9 + 5713*n^8 - 47417*n^7 + 250708*n^6 - 883176*n^5 + 2104831*n^4 - 3368071*n^3 + 3489712*n^2 - 2133004*n + 587808)*a(n) = 2*(n-1)*n*(n+1)*(24*n^10 - 756*n^9 + 10262*n^8 - 78647*n^7 + 374743*n^6 - 1154043*n^5 + 2323495*n^4 - 3057578*n^3 + 2632172*n^2 - 1456776*n + 412560)*a(n-1) + 4*(n-1)*n*(12*n^11 - 384*n^10 + 5377*n^9 - 43234*n^8 + 219811*n^7 - 731024*n^6 + 1576767*n^5 - 2055172*n^4 + 1195025*n^3 + 527398*n^2 - 1223056*n + 534240)*a(n-2) - 2*(72*n^13 - 2484*n^12 + 37950*n^11 - 339019*n^10 + 1971954*n^9 - 7887993*n^8 + 22425262*n^7 - 46437513*n^6 + 71577166*n^5 - 83189763*n^4 + 71509420*n^3 - 41716412*n^2 + 13543200*n - 1451520)*a(n-3) - 4*(n-1)*n*(2*n - 7)*(24*n^10 - 756*n^9 + 10262*n^8 - 78647*n^7 + 374743*n^6 - 1154043*n^5 + 2323495*n^4 - 3057578*n^3 + 2632172*n^2 - 1456776*n + 412560)*a(n-4) - 8*(n-1)*(2*n - 9)*(12*n^11 - 384*n^10 + 5377*n^9 - 43234*n^8 + 219811*n^7 - 731024*n^6 + 1576767*n^5 - 2055172*n^4 + 1195025*n^3 + 527398*n^2 - 1223056*n + 534240)*a(n-5) + 16*(n-6)*(2*n - 11)*(2*n - 9)*(12*n^10 - 276*n^9 + 2689*n^8 - 14529*n^7 + 48009*n^6 - 101629*n^5 + 142510*n^4 - 137838*n^3 + 93836*n^2 - 39760*n + 6720)*a(n-6). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 4^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

Name edited by Andrew Howroyd, Nov 20 2017

A005035 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

1, 1, 4, 13, 64, 315, 1727, 9658, 55657, 325390, 1929160, 11555172, 69840032, 425318971, 2607388905, 16077392564, 99646239355, 620439153165, 3879069845640, 24342884609625, 153279112388352, 968123122592340, 6131992590993204, 38940057166651848

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..200
F. Harary, E. M. Palmer, R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

Column k=4 of A295259.

u[n_, k_, r_] := r*Binomial[(k-1)*n + r, n]/((k-1)*n + r);
F[n_, k_] := DivisorSum[GCD[n-1, k], EulerPhi[#]*u[(n-1)/#, k, k/#]&]/k;
T[n_, k_] := (F[n, k] + If[OddQ[k], If[OddQ[n], u[(n-1)/2, k, (k-1)/2], u[n/2-1, k, k-1]], If[OddQ[n], u[(n-1)/2, k, k/2+1], u[n/2-1, k, k]]])/2;
a[n_] := T[n, 4];
Array[a, 24] (* Jean-François Alcover, Jul 02 2018, after Andrew Howroyd *)

More terms from Sean A. Irvine, Mar 11 2016

Name edited by Andrew Howroyd, Nov 20 2017

A005039 Number of nonequivalent dissections of a polygon into n pentagons by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

1, 1, 4, 22, 147, 1074, 8216, 64798, 521900, 4272967, 35447724, 297308810, 2516830890, 21476307960, 184530904560, 1595190209002, 13863857007924, 121067796450692, 1061770618201680, 9347742325179544, 82584606893075739

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
F. Harary, E. M. Palmer, R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.

Column k=5 of A295259.

Cf. A002293, A005037 (no mirror-image symmetries), A003446 (triangles), A005035 (quadrilaterals).

Rest[CoefficientList[Series[x*(HypergeometricPFQ[{1/4, 1/2, 3/4}, {2/3, 4/3}, (256/27)*x]^5 + 4*HypergeometricPFQ[{1/4, 1/2, 3/4}, {2/3, 4/3}, (256/27)*x^5] + 5*HypergeometricPFQ[{1/4, 1/2, 3/4}, {2/3, 4/3}, (256/27)*x^2]^2 + 5*x*HypergeometricPFQ[{1/4, 1/2, 3/4}, {2/3, 4/3}, (256/27)*x^2]^4)/10, {x, 0, 25}], x]] (* Vaclav Kotesovec, Mar 13 2016 *)

G.f.: (1/10)*x*(u^5(x) + 4*u(x^5) + 5*u^2(x^2) + 5*x*u^4(x^2)) where u(x) is the g.f. for A002293. - Sean A. Irvine, Mar 12 2016

a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Mar 13 2016

More terms from Sean A. Irvine, Mar 12 2016

Name edited by Andrew Howroyd, Nov 20 2017

cp's OEIS Frontend

A295260 Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals up to rotation and reflection (k >= 3).

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A295224 Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals up to rotation (k >= 3).

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A295222 Array read by antidiagonals: T(n,k) is the number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell up to rotation (k >= 3).

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A003456 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation and reflection.

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A370060 Array read by antidiagonals: T(n,k) is the number of achiral dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell, n >= 1, k >= 3.

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A003446 Number of nonequivalent dissections of a polygon into n triangles by nonintersecting diagonals rooted at a cell up to rotation and reflection.

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A005035 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals rooted at a cell up to rotation and reflection.

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