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 A070914 #44 Jan 05 2025 19:51:37
%S A070914 1,1,1,1,1,1,1,1,2,1,1,1,3,5,1,1,1,4,12,14,1,1,1,5,22,55,42,1,1,1,6,
%T A070914 35,140,273,132,1,1,1,7,51,285,969,1428,429,1,1,1,8,70,506,2530,7084,
%U A070914 7752,1430,1,1,1,9,92,819,5481,23751,53820,43263,4862,1,1,1,10,117,1240
%N A070914 Array read by antidiagonals giving number of paths up and left from (0,0) to (n,kn) where x/y <= k for all intermediate points.
%C A070914 Also related to dissections of polygons and enumeration of trees.
%C A070914 Number of dissections of a polygon into n (k+2)-gons by nonintersecting diagonals. All dissections are counted separately. See A295260 for nonequivalent solutions up to rotation and reflection. - _Andrew Howroyd_, Nov 20 2017
%C A070914 Number of rooted polyominoes composed of n (k+2)-gonal cells of the hyperbolic (Euclidean for k=0) regular tiling with Schläfli symbol {k+2,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. For k>0, a stereographic projection of the {k+2,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - _Robert A. Russell_, Jan 27 2024
%H A070914 Alois P. Heinz, <a href="/A070914/b070914.txt">Antidiagonals n = 0..140, flattened</a>
%H A070914 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A070914 Peter Hilton and Jean Pedersen, <a href="http://www.math.uakron.edu/~cossey/636papers/hilton%20and%20pedersen.pdf">Catalan Numbers, Their Generalization, and Their Uses</a>, The Mathematical Intelligencer, March 1991, Volume 13, Issue 2, pp. 64-75.
%H A070914 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.
%F A070914 T(n, k) = binomial(n*(k+1), n)/(n*k+1) = A071201(n, k*n) = A071201(n, k*n+1) = A071202(n, k*n+1) = A062993(n+k-1, k-1).
%F A070914 If P(k,x) = Sum_{n>=0} T(n,k)*x^n is the g.f. of column k (k>=0), then P(k,x) = exp(1/(k+1)*(Sum_{j>0} (1/j)*binomial((k+1)*j,j)*x^j)). - _Werner Schulte_, Oct 13 2015
%e A070914 Rows start:
%e A070914 ===========================================================
%e A070914 n\k| 0 1 2 3 4 5 6
%e A070914 ---|-------------------------------------------------------
%e A070914 0 | 1, 1, 1, 1, 1, 1, 1 ...
%e A070914 1 | 1, 1, 1, 1, 1, 1, 1 ...
%e A070914 2 | 1, 2, 3, 4, 5, 6, 7 ...
%e A070914 3 | 1, 5, 12, 22, 35, 51, 70 ...
%e A070914 4 | 1, 14, 55, 140, 285, 506, 819 ...
%e A070914 5 | 1, 42, 273, 969, 2530, 5481, 10472 ...
%e A070914 6 | 1, 132, 1428, 7084, 23751, 62832, 141778 ...
%e A070914 7 | 1, 429, 7752, 53820, 231880, 749398, 1997688 ...
%e A070914 8 | 1, 1430, 43263, 420732, 2330445, 9203634, 28989675 ...
%e A070914 ...
%p A070914 A:= (n, k)-> binomial((k+1)*n, n)/(k*n+1):
%p A070914 seq(seq(A(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Mar 25 2015
%t A070914 T[n_, k_] = Binomial[n(k+1), n]/(k*n+1); Flatten[Table[T[n-k, k], {n, 0, 9}, {k, n, 0, -1}]] (* _Jean-François Alcover_, Apr 08 2016 *)
%o A070914 (PARI) T(n, k) = binomial(n*(k+1), n)/(n*k+1); \\ _Andrew Howroyd_, Nov 20 2017
%Y A070914 Rows include A000012 (twice), A000027, A000326.
%Y A070914 Columns include A000012, A000108 (Catalan), A001764, A002293, A002294, A002295, A002296, A007556, A062994, A059968.
%Y A070914 Reflected version of A062993 (which is the main entry).
%Y A070914 Cf. A295260.
%Y A070914 Polyominoes: A295224 (oriented), A295260 (unoriented).
%K A070914 nonn,tabl
%O A070914 0,9
%A A070914 _Henry Bottomley_, May 20 2002