This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A295260 #17 Jan 28 2024 09:21:15
%S A295260 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,
%T A295260 16,68,194,261,82,1,1,4,21,112,483,1196,1243,228,1,1,5,27,183,1020,
%U A295260 3946,8196,6257,733,1,1,5,33,266,1918,10222,34485,58140,32721,2282
%N 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).
%C A295260 The polygon prior to dissection will have n*(k-2)+2 sides.
%C A295260 In the Harary, Palmer and Read reference these are the sequences called h.
%C A295260 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
%H A295260 Andrew Howroyd, <a href="/A295260/b295260.txt">Table of n, a(n) for n = 1..1275</a>
%H A295260 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A295260 F. Harary, E. M. Palmer and R. C. Read, <a href="http://dx.doi.org/10.1016/0012-365X(75)90041-2">On the cell-growth problem for arbitrary polygons</a>, Discr. Math. 11 (1975), 371-389.
%H A295260 E. Krasko, A. Omelchenko, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i1p17">Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees</a>, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
%H A295260 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">Fuss-Catalan number</a>
%F A295260 T(n,k) ~ A295222(n,k)/(2*n) for fixed k.
%e A295260 Array begins:
%e A295260 ===================================================
%e A295260 n\k| 3 4 5 6 7 8
%e A295260 ---|-----------------------------------------------
%e A295260 1 | 1 1 1 1 1 1 ...
%e A295260 2 | 1 1 1 1 1 1 ...
%e A295260 3 | 1 2 2 3 3 4 ...
%e A295260 4 | 3 5 8 12 16 21 ...
%e A295260 5 | 4 16 33 68 112 183 ...
%e A295260 6 | 12 60 194 483 1020 1918 ...
%e A295260 7 | 27 261 1196 3946 10222 22908 ...
%e A295260 8 | 82 1243 8196 34485 109947 290511 ...
%e A295260 9 | 228 6257 58140 315810 1230840 3844688 ...
%e A295260 10 | 733 32721 427975 2984570 14218671 52454248 ...
%e A295260 ...
%t A295260 u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
%t A295260 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;
%t A295260 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_ *)
%o A295260 (PARI) \\ here u is Fuss-Catalan sequence with p = k+1.
%o A295260 u(n,k,r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
%o A295260 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}
%o A295260 for(n=1, 10, for(k=3, 8, print1(T(n, k), ", ")); print);
%Y A295260 Columns k=3..7 are A000207, A005036, A005040, A004127, A005419.
%Y A295260 Cf. A033282, A295222, A295259.
%Y A295260 Polyominoes: A295224 (oriented), A070914 (rooted).
%K A295260 nonn,tabl
%O A295260 1,9
%A A295260 _Andrew Howroyd_, Nov 18 2017