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 A255908 #31 Sep 08 2022 08:46:11
%S A255908 2,4,8,8,32,48,16,128,288,384,32,512,1728,3072,3840,64,2048,10368,
%T A255908 24576,38400,46080,128,8192,62208,196608,384000,552960,645120,256,
%U A255908 32768,373248,1572864,3840000,6635520,9031680,10321920,512,131072,2239488,12582912,38400000,79626240,126443520,165150720,185794560,1024,524288,13436928,100663296,384000000,955514880,1770209280,2642411520,3344302080,3715891200
%N A255908 Triangle read by rows: T(n,L) = number of rho-labeled graphs with n edges whose labeling is bipartite with boundary value L.
%C A255908 A graph with n edges is rho-labeled if there exists a one-to-one mapping from its vertex set to {0,1,...,2n} such that every edge receives as label the absolute difference of its end-vertices and the edge labels are x_1, x_2, ..., x_n where x_i = i or x_i = 2n + 1 - i. A rho-labeling of a bipartite graph is said to be bipartite when the labels of one stable set are smaller than the labels on the other stable set. The largest of the smaller vertex labels is its boundary value.
%C A255908 From _Robert G. Wilson v_, Jul 05 2015: (Start)
%C A255908 The columns:
%C A255908 T(n, 0) = 2^n,
%C A255908 T(n, 1) = 2^(2n-1),
%C A255908 T(n, 2) = 2^(n+1)*3^(n-2),
%C A255908 T(n, 3) = 3*2^(3n-5),
%C A255908 T(n, 4) = 3*2^(n+3)*5^(n-4),
%C A255908 T(n, 5) = 5*2^(2n-2)*3^(n-4), etc.
%C A255908 The diagonals:
%C A255908 the main, T(n, n-1) = 2^n*n*(n-1!) = 2*A002866,
%C A255908 the second diagonal, T(n, n-2) = 2^n*(n-1)^2*(n-2)! = 4*A014479,
%C A255908 the third diagonal, T(n, n-3) = 2^n*(n-2)^3*(n-3)!,
%C A255908 the k_th diagonal, T(n, n-k) = 2^n*(n-k)^k*(n-k)!, etc.
%C A255908 ... (End)
%H A255908 Joseph A. Gallian, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6">A dynamic survey of graph labeling</a>, Elec. J. Combin., (2014), #DS6.
%F A255908 For n>=1, 0<=L<=n-1, T(n,L)=(2^n)*(L!)*(L+1)^(n-L).
%e A255908 For n=5 and L=1, T(5,1)=(2^5)*(1!)*(1+1)^(5-1)=512.
%e A255908 For n=9 and L=3, T(9,3)=12582912.
%e A255908 Triangle, T, begins:
%e A255908 -----------------------------------------------------------------------------
%e A255908 n\L | 0 1 2 3 4 5 6
%e A255908 ----|------------------------------------------------------------------------
%e A255908 1 | 2;
%e A255908 2 | 4, 8;
%e A255908 3 | 8, 32, 48;
%e A255908 4 | 16, 128, 288, 384;
%e A255908 5 | 32, 512, 1728, 3072, 3840;
%e A255908 6 | 64, 2048, 10368, 24576, 38400, 46080;
%e A255908 7 | 128, 8192, 62208, 196608, 384000, 552960, 645120;
%e A255908 8 | 256, 32768, 373248, 1572864, 3840000, 6635520, 9031680, ...
%e A255908 ...
%e A255908 For n=2 and L=1, T(2,1)=8, because: the bipartite graph <{v1,v2,v3},{x1=v1v2,x2=v1v3}> has rho-labelings (2,1,3),(2,1,4) with L=1 on the stable set {x2} and rho-labelings (1,2,0),(0,4,1) with L=1 on the stable set {x1,x3}; the bipartite graph <{v1,v2,v3,v4},{x1=v1v2,x2=v3v4}> has rho-labeling (0,4,1,3),(1,2,0,3) with L=1 on the stable set {v1,v3} and rho-labeling (4,0,3,1),(2,1,3,0) with L=1 on the stable set {v2,v4}. - _Danny Rorabaugh_, Apr 03 2015
%t A255908 t[n_, l_] := 2^n*l!(l+1)^(n-l); Table[ t[n, l], {n, 8}, {l, 0, n-1}] // Flatten (* _Robert G. Wilson v_, Jul 05 2015 *)
%o A255908 (Magma) [2^n*Factorial(l)*(l+1)^(n-l): l in [0..n-1], n in [1..10]]; // _Bruno Berselli_, Aug 05 2015
%K A255908 easy,nonn,tabl
%O A255908 1,1
%A A255908 _Christian Barrientos_ and _Sarah Minion_, Mar 10 2015