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 A111516 #43 Dec 29 2023 13:19:26
%S A111516 1,1,1,1,3,4,1,7,12,18,1,15,32,56,88,1,31,80,160,280,450,1,63,192,432,
%T A111516 832,1452,2364,1,127,448,1120,2352,4424,7700,12642,1,255,1024,2816,
%U A111516 6400,12896,23872,41456,68464,1,511,2304,6912,16896,36288,71136,130176,225648,374274
%N A111516 Triangle G(n,k) read by rows: number of order-preserving partial transformations (of an n-element totally ordered set) of waist k (waist(alpha) = max(Im(alpha))).
%H A111516 A. Laradji and A. Umar, <a href="https://doi.org/10.1016/j.jalgebra.2003.10.023">Combinatorial results for semigroups of order-preserving partial transformations</a>, Journal of Algebra, 278, (2004), 342-359.
%H A111516 A. Laradji and A. Umar, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial results for semigroups of order-decreasing partial transformations</a>, J. Integer Seq., 7 (2004), 04.3.8.
%F A111516 G(n,k) = Sum_{j=1..n} C(n,j)*C(k+j-2,j-1) for 1 <= k <= n. [Corrected by _Petros Hadjicostas_, Feb 13 2021]
%F A111516 G(n,k) = 2*G(n-1,k) - G(n-1,k-1) + G(n,k-1) for n >= 2 and 1 <= k <= n-1 with initial conditions G(n,0) = 1 for n >= 0, G(n,1) = 2^n - 1 for n >= 1, and G(n,n) = A050146(n) for n >= 2.
%F A111516 Sum_{k=0..n} G(n,k) = A123164(n).
%F A111516 From _Petros Hadjicostas_, Feb 13 2021: (Start)
%F A111516 G(n,k) = A055807(n+k,k) for 0 <= k <= n.
%F A111516 Bivariate o.g.f.: Sum_{n,k>=0} G(n,k)*x^n*y^k = ((2 - 2*y - 2*x*y + x*y^2) + 2*x*(y - 1)/(1 - x) + x*y*(2 - 3*y - 2*x*y + x*y^2)/sqrt(1 - 6*x*y + x^2*y^2))/(2*(1 - 2*x - y + x*y)). (End)
%e A111516 Triangle G(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A111516 1;
%e A111516 1, 1;
%e A111516 1, 3, 4;
%e A111516 1, 7, 12, 18;
%e A111516 1, 15, 32, 56, 88;
%e A111516 1, 31, 80, 160, 280, 450;
%e A111516 1, 63, 192, 432, 832, 1452, 2364;
%e A111516 ...
%e A111516 G(2,2) = 4 because there are exactly 4 order-preserving partial transformations (on a 2-element chain) of waist 2, namely: (1)->(2), (2)->(2), (1,2)->(1,2), and (1,2)->(2,2) - the mappings are coordinate-wise.
%Y A111516 Cf. A050146 (main diagonal), A055807, A123164 (row sums).
%K A111516 nonn,tabl
%O A111516 0,5
%A A111516 _Abdullahi Umar_, Aug 25 2008
%E A111516 G(7,5) corrected by and more terms from _Petros Hadjicostas_, Feb 13 2021