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
Examples
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 ... ...
Links
- 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
Crossrefs
Programs
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Mathematica
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 *) -
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); -
Python
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
Formula
T(n,k) ~ A295222(n,k)/2 for fixed k.
Comments