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 A295224 #25 Nov 24 2024 00:07:48
%S A295224 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,
%T A295224 4,26,118,366,492,150,1,1,4,35,203,931,2340,2431,442,1,1,5,46,332,
%U A295224 1989,7756,16252,12371,1424,1,1,5,57,494,3766,20254,68685,115940,65169,4522
%N 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).
%C A295224 The polygon prior to dissection will have n*(k-2)+2 sides.
%C A295224 In the Harary, Palmer and Read reference these are the sequences called H.
%C A295224 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
%H A295224 Andrew Howroyd, <a href="/A295224/b295224.txt">Table of n, a(n) for n = 1..1275</a>
%H A295224 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A295224 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 A295224 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 A295224 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">Fuss-Catalan number</a>
%F A295224 T(n,k) ~ A295222(n,k)/n for fixed k.
%e A295224 Array begins:
%e A295224 =====================================================
%e A295224 n\k| 3 4 5 6 7 8
%e A295224 ---|-------------------------------------------------
%e A295224 1 | 1 1 1 1 1 1 ...
%e A295224 2 | 1 1 1 1 1 1 ...
%e A295224 3 | 1 2 2 3 3 4 ...
%e A295224 4 | 4 7 12 19 26 35 ...
%e A295224 5 | 6 25 57 118 203 332 ...
%e A295224 6 | 19 108 366 931 1989 3766 ...
%e A295224 7 | 49 492 2340 7756 20254 45448 ...
%e A295224 8 | 150 2431 16252 68685 219388 580203 ...
%e A295224 9 | 442 12371 115940 630465 2459730 7684881 ...
%e A295224 10 | 1424 65169 854981 5966610 28431861 104898024 ...
%e A295224 ...
%t A295224 u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
%t A295224 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;
%t A295224 Table[T[n - k + 1, k], {n, 1, 13}, {k, n, 3, -1}] // Flatten (* _Jean-François Alcover_, Nov 21 2017, after _Andrew Howroyd_ *)
%o A295224 (PARI) \\ here u is Fuss-Catalan sequence with p = k+1.
%o A295224 u(n, k, r)={r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
%o A295224 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;
%o A295224 for(n=1, 10, for(k=3, 8, print1(T(n, k), ", ")); print);
%o A295224 (Python)
%o A295224 from sympy import binomial, gcd, totient, divisors
%o A295224 def u(n, k, r): return r*binomial((k - 1)*n + r, n)//((k - 1)*n + r)
%o A295224 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
%o A295224 for n in range(1, 11): print([T(n, k) for k in range(3, 9)]) # _Indranil Ghosh_, Dec 13 2017, after PARI code
%Y A295224 Columns k=3..6 are A001683(n+2), A005034, A005038, A221184(n-1).
%Y A295224 Cf. A033282, A070914, A295222, A295259, A295260.
%Y A295224 Polyominoes: A295260 (unoriented), A070914 (rooted).
%K A295224 nonn,tabl
%O A295224 1,9
%A A295224 _Andrew Howroyd_, Nov 17 2017