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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). 15
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

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.

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.

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

FORMULA

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

EXAMPLE

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 ...

  ...

MATHEMATICA

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 *)

PROG

(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)}

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);

CROSSREFS

Columns k=3..7 are A000207, A005036, A005040, A004127, A005419.

Cf. A033282, A070914, A295222, A295224, A295259.

Sequence in context: A249128 A046226 A054722 * A256187 A251045 A271566

Adjacent sequences:  A295257 A295258 A295259 * A295261 A295262 A295263

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Nov 18 2017

STATUS

approved

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Last modified February 18 17:08 EST 2018. Contains 299325 sequences. (Running on oeis4.)