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 A192933 #74 Jan 03 2024 07:39:19
%S A192933 1,1,2,2,6,12,4,16,44,88,8,40,136,360,720,16,96,384,1216,3152,6304,32,
%T A192933 224,1024,3712,11296,28896,57792,64,512,2624,10624,36416,108032,
%U A192933 273856,547712,128,1152,6528,29056,109696,362624,1056896,2661504,5323008,256,2560,15872,76800,314880,1135616,3659776,10528768,26380544,52761088
%N A192933 Triangle read by rows: T(n,k) = Sum_{i <= n, j <= k, (i,j) <> (n,k)} T(i,j), starting with T(1,1) = 1, for n >= 1 and 1 <= k <= n.
%C A192933 The outer diagonal is A059435.
%C A192933 The second outer diagonal is A090442.
%C A192933 The third outer diagonal is essentially 2*A068766.
%C A192933 The first column is A011782.
%C A192933 The second column is essentially A057711 (not considering its first two terms).
%C A192933 The second column is essentially A129952 (not considering its first two terms).
%C A192933 The second column is essentially 2*A001792.
%C A192933 The differences between the terms of the second column is essentially 2*A045623.
%C A192933 The third column is essentially 4*A084266.
%C A192933 The cumulative sums of the third column are essentially 4*A176027.
%C A192933 T(n,k) = 0 for n < k. If this overriding constraint is not applied, you get A059576. - _Franklin T. Adams-Watters_, Jul 24 2011
%C A192933 For n >= 2 and 1 <= k <= n, T(n,k) is the number of bimonotone subdivisions of a 2-row grid with n points on the first row and k points on the second row (with the lower left point of the grid being the origin). A bimonotone subdivision of a convex polygon (the convex hull of the grid) is one where the internal dividing lines have nonnegative (including infinite) slopes. See Robeva and Sun (2020). - _Petros Hadjicostas_ and _Michel Marcus_, Jul 15 2020
%H A192933 Andrea Raffetti, <a href="/A192933/b192933.txt">Rows n = 1..13 of triangle</a>
%H A192933 Elina Robeva and Melinda Sun, <a href="https://arxiv.org/abs/2007.00877">Bimonotone Subdivisions of Point Configurations in the Plane</a>, arXiv:2007.00877 [math.CO], 2020.
%F A192933 T(n,1) = 2^(n-2) for n >= 2.
%F A192933 T(n,2) = n*2^(n-2) for n >= 2.
%F A192933 T(n,3) = 2^(n-2)*((n-k+1)^2 + 7*(n-k+1) + 4)/2 = 2^(n-3)*(n^2 + 3*n - 6) for k = 3 and n >= 3.
%F A192933 In general: For 1 <= k <= n with (n,k) <> 1,
%F A192933 T(n,k) = 2^(n-2)*Sum_{i=0..k-1} c(k,i)*(n-k+1)^(k-1-i)/(k-1)! and
%F A192933 T(n,k) = 2^(n-2)*Sum_{j=0..k-1} c(k,k-1-j)*(n-k+1)^j/(k-1)!
%F A192933 with c(k,i) being specific coefficients. Below are the first values for c(k,i):
%F A192933 1;
%F A192933 1, 1;
%F A192933 1, 7, 4;
%F A192933 1, 18, 77, 36;
%F A192933 1, 34, 359, 1238, 528,
%F A192933 1, 55, 1065, 8705, 26654, 10800;
%F A192933 ... [Formula corrected by _Petros Hadjicostas_, Jul 15 2020]
%F A192933 The diagonal of this triangle for c(k,i) divided by (k-1)! (except for the first term) is equal to the Shroeder number sequence A006318(k-1).
%F A192933 From _Petros Hadjicostas_ and _Michel Marcus_, Jul 15 2020: (Start)
%F A192933 T(n,1) = 2^(n-2) for n >= 2; T(n,k) = 2*(T(n,k-1) + T(n-1,k) - T(n-1,k-1)) for n > k >= 2; T(n,n) = 2*T(n,n-1) for n = k >= 2; and T(n,k) = 0 for 1 <= n < k. [Robeva and Sun (2020)] (They do not specify T(1,1) explicitly since they do not care about subdivisions of a degenerate polygon with only one side.)
%F A192933 T(n,k) = (2^(n-2)/(k-1)!) * P_k(n) = (2^(n-2)/(k-1)!) * Sum_{j=1..k} A336245(k,j)*n^(k-j) for n >= k >= 1 with (n,k) <> (1,1), where P_k(n) is some polynomial with integer coefficients of degree k-1. [Robeva and Sun (2020)]
%F A192933 A336245(k,j) = Sum_{s=0..j-1} c(k,s) * binomial(k-1-s, k-j) * (1-k)^(j-1-s) for 1 <= j <= k, in terms of the above coefficients c(k,i).
%F A192933 So c(k,s) = Sum_{j=1..s+1} A336245(k,j) * binomial(k-j, k-s-1) * (k-1)^(s+1-j) for k >= 1 and 0 <= s <= k-1, obtained by inverting the binomial transform.
%F A192933 Bivariate o.g.f.: x*y*(1 - x)*(1 - 2*y*g(2*x*y))/(1 - 2*x - 2*y + 2*x*y), where g(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = g.f. of A001003.
%F A192933 Letting y = 1 in the above joint o.g.f., we get the o.g.f. of the row sums: x*(1-x)*(2*g(2*x) - 1). It can then be easily proved that
%F A192933 Sum_{k=1..n} T(n,k) = 2^n*A001003(n-1) - 2^(n-1)*A001003(n-2) for n >= 3. (End)
%e A192933 Triangle (with rows n >= 1 and columns k = 1..n) begins:
%e A192933 1;
%e A192933 1, 2;
%e A192933 2, 6, 12;
%e A192933 4, 16, 44, 88;
%e A192933 8, 40, 136, 360, 720;
%e A192933 16, 96, 384, 1216, 3152, 6304;
%e A192933 32, 224, 1024, 3712, 11296, 28896, 57792;
%e A192933 64, 512, 2624, 10624, 36416, 108032, 273856, 547712;
%e A192933 ...
%e A192933 Example: T(4,3) = 44 = 1 + 1 + 2 + 2 + 6 + 12 + 4 + 16.
%e A192933 From _Petros Hadjicostas_, Jul 15 2020: (Start)
%e A192933 Consider the following 2-row grid with n = 3 points at the top and k = 2 points at the bottom:
%e A192933 A B C
%e A192933 *--*--*
%e A192933 | /
%e A192933 | /
%e A192933 *--*
%e A192933 D E
%e A192933 The sets of the dividing internal lines of the T(3,2) = 6 bimonotone subdivisions of the above 2-row grid are as follows: { }, {DC}, {DB}, {EB}, {DB, DC}, and {DB, EB}. We exclude subdivisions {EA} and {EA, EB} because they have at least one dividing line with a negative slope. (End)
%o A192933 (PARI) lista(nn) = {my(T=matrix(nn, nn)); T[1,1] = 1; for (n=2, nn, for (k=1, n, T[n,k] = sum(i=1, n, sum(j=1, k, if ((i!=n) || (j!=k), T[i,j]))););); vector(nn, k, vector(k, i, T[k, i]));} \\ _Michel Marcus_, Mar 18 2020
%Y A192933 Cf. A001003, A001792, A006318, A011782, A045623, A057711, A059435, A059576, A068766, A084266, A090442, A129952, A166444, A176027, A336245.
%K A192933 nonn,tabl
%O A192933 1,3
%A A192933 _Andrea Raffetti_, Jul 13 2011
%E A192933 Offset changed by _Andrew Howroyd_, Dec 31 2017
%E A192933 Name edited by _Petros Hadjicostas_, Jul 15 2020