A003446 -id:A003446 - cp's OEIS frontend

A295259 Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell up to rotation and reflection (k >= 3).

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 4, 13, 16, 1, 1, 6, 22, 64, 52, 1, 1, 6, 35, 147, 315, 170, 1, 1, 8, 49, 302, 1074, 1727, 579, 1, 1, 8, 67, 518, 2763, 8216, 9658, 1996, 1, 1, 10, 87, 843, 5916, 27168, 64798, 55657, 7021

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 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 |    2      4       4        6         6         8 ...
   4 |    6     13      22       35        49        67 ...
   5 |   16     64     147      302       518       843 ...
   6 |   52    315    1074     2763      5916     11235 ...
   7 |  170   1727    8216    27168     70984    159180 ...
   8 |  579   9658   64798   274360    876790   2319678 ...
   9 | 1996  55657  521900  2837208  11069760  34582800 ...
  10 | 7021 325390 4272967 29828330 142148343 524470485 ...
  ...

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 A003446, A005035, A005039.

Cf. A033282, A070914, A295222, A295224, A295260.

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;
Table[T[n-k-1, k], {n, 1, 14}, {k, n-2, 3, -1}] // Flatten (* Jean-François Alcover, Jan 19 2018, translated from PARI *)

\\ 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)}
F(n,k) = {sumdiv(gcd(n-1,k), d, eulerphi(d)*u((n-1)/d,k,k/d))/k}
T(n,k) = {(F(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)) ))/2;}
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 F(n, k): return sum([totient(d)*u((n - 1)//d, k, k//d) for d in divisors(gcd(n - 1, k))])//k
def T(n, k): return (F(n, k) + ((u((n - 1)//2, k, (k - 1)//2) if n%2 else u(n//2 - 1, k, k - 1)) if k%2 else (u((n - 1)//2, k, k//2 + 1) if n%2 else u(n//2 - 1, k, k))))//2
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)/2 for fixed k.

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

A000908 Atom-rooted polyenoids with n edges with symmetry class C_s.

Original entry on oeis.org

0, 0, 1, 4, 14, 47, 164, 565, 1982, 6977, 24850, 89082, 321855, 1169853, 4276923, 15713799, 57998270, 214934984, 799473752, 2983682702, 11169374372, 41929478873, 157807392886, 595340271682, 2250901007539, 8527699269192, 32369066434276

Offset: 0

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

nonn

B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, Isomers of polyenes attached to benzene, Croatica Chemica Acta, 68 (1995), 63-73, C(x).
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. A000912, A000913, A000935, A000936, A000941, A000942, A000947, A000948, A000953, A003446, A063786.

U0 := (1-sqrt(1-4*x))/2/x ;
V0 := 1+x*subs(x=x^2,U0) ;
C := ( subs(x=x^2,U0)^3 -3*subs(x=x^4,U0)*subs(x=x^2,V0) -subs(x=x^6,U0) +3*subs(x=x^6,V0) )/6 ; # (19)
taylor(%,x=0,60) ;
L := gfun[seriestolist](%) ;
seq(op(2*i+1,L),i=0..(nops(L)-1)/2) ; # R. J. Mathar, Jul 26 2019

u0[x_] := (1 - Sqrt[1 - 4 x])/(2 x); v0[x_] := 1 + x u0[x^2];
gf = Simplify[(u0[x]^3 - 3 u0[x^2] v0[x] - u0[x^3] + 3 v0[x^3])/6]
CoefficientList[gf + O[x]^30, x] (* Andrey Zabolotskiy, Feb 08 2023 *)

a(n) = A003446(n+1) - u((n-3)/6) - (u(n/3) - u((n-3)/6))/2 - (u(n/2) + (u((n+1)/2) - u((n-3)/6))) for n > 0 where u(n) = binomial(2*n, n)/(n+1) if n is an integer and 0 otherwise. - Sean A. Irvine, Oct 05 2015

More terms from Sean A. Irvine, Oct 05 2015

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A295259 Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell up to rotation and reflection (k >= 3).

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

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A000908 Atom-rooted polyenoids with n edges with symmetry class C_s.

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