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 A137854 #12 Apr 03 2022 01:31:39 %S A137854 1,1,1,1,2,1,1,4,4,1,1,8,11,8,1,1,16,28,28,16,1,1,32,71,87,71,32,1,1, %T A137854 64,184,266,266,184,64,1,1,128,491,823,952,823,491,128,1,1,256,1348, %U A137854 2598,3381,381,2598,1348,2561 %N A137854 Triangle generated from an array: A008277 * A008277(transform). %C A137854 Row sums = A000995 such that row 1 = A000995(3) = 1. %C A137854 This array is the product of the lower triangular Stirling matrix and its transpose, which explains why the array is symmetric. - _David Callan_, Dec 02 2011 %C A137854 In the triangle, T(n,k) is the number of permutations of [n+1] that avoid both dashed patterns 1-23 and 3-12, start with an ascent, and have first entry k. For example, T(4,2)=4 counts 23154, 24153, 24315, 25431. - _David Callan_, Dec 02 2011 %F A137854 Triangle read by rows = antidiagonals of an array formed by A008277 * A008277(transform), where A008277 = the Stirling number of the second kind triangle. %e A137854 First few rows of the array: %e A137854 1, 1, 1, 1, 1, 1, ... %e A137854 1, 2, 4, 8, 16, 32, ... %e A137854 1, 4, 11, 28, 71, 184, ... %e A137854 1, 8, 28, 87, 266, 823, ... %e A137854 1, 16, 71, 266, 952, 3381, ... %e A137854 ... %e A137854 First few rows of the triangle: %e A137854 1; %e A137854 1, 1; %e A137854 1, 2, 1; %e A137854 1, 4, 4, 1; %e A137854 1, 8, 11, 8, 1; %e A137854 1, 16, 28, 28, 16, 1; %e A137854 1, 32, 71, 87, 71, 32, 1; %e A137854 1, 64, 184, 266, 266, 184, 64, 1; %e A137854 1, 128, 491, 823, 952, 823, 491, 128, 1; %e A137854 ... %Y A137854 Cf. A000995, A008277. %K A137854 nonn,tabl %O A137854 1,5 %A A137854 _Gary W. Adamson_, Feb 15 2008