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 A115154 #15 Aug 06 2024 09:01:25
%S A115154 1,1,4,1,13,25,1,40,115,190,1,121,466,1036,1606,1,364,1762,4870,9688,
%T A115154 14506,1,1093,6379,20989,50053,93571,137089,1,3280,22417,85384,235543,
%U A115154 516256,927523,1338790,1,9841,77092,333244,1039873,2588641,5371210
%N A115154 Triangle of numbers related to the generalized Catalan sequence C(3;n+1) = A064063(n+1), n>=0.
%C A115154 This triangle, called Y(3,1), appears in the totally asymmetric exclusion process for the (unphysical) values alpha=3, beta=1. See the Derrida et al. refs. given under A064094, where the triangle entries are called Y_{N,K} for given alpha and beta.
%C A115154 The main diagonal (M=1) gives the generalized Catalan sequence C(3,n+1):=A064063(n+1).
%C A115154 The diagonal sequences give A064063(n+1), A115188-A115192 for n+1>= M=1,..,6.
%H A115154 B. Derrida, E. Domany and D. Mukamel, <a href="https://dx.doi.org/10.1007/BF01050430">An exact solution of a one-dimensional asymmetric exclusion model with open boundaries</a>, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
%H A115154 Wolfdieter Lang, <a href="/A115154/a115154.txt">First 10 rows</a>.
%F A115154 a(n,n+1)=A064063(n+1) (main diagonal with M=1); a(n,n-M+2)= a(n,n-M+1) + 3*a(n-1,n-M+2), M>=2; a(n,1)=1; n>=0.
%F A115154 G.f. for diagonal sequence M=1: GY(1,x):=(3*c(3*x)-1)/(2+x) with c(x) the o.g.f. of A000108 (Catalan); for M=2: GY(2,x)=(1-3*x)*GY(1,x)-1; for M>=3: GY(M,x)= GY(M-1,x) - 3*x*GY(M-2,x) + 2*x^(M-2).
%F A115154 G.f. for diagonal sequence M (solution to the above given recurrence): GY(M,x)= (x^(M-1)/(1+x))*( 3^(M+1)*x*(p(M,3*x)-(3*x)*p(M+1,3*x)*c(3*x))+1), with c(x) g.f. of A000108 (Catalan) and p(n,x):= -((1/sqrt(x))^(n+1))*S(n-1,1/sqrt(x)) with Chebyshev's S(n,x) polynomials given in A049310.
%e A115154 Triangle begins:
%e A115154 1;
%e A115154 1, 4;
%e A115154 1, 13, 25;
%e A115154 1, 40, 115, 190;
%e A115154 1, 121, 466, 1036, 1606;
%e A115154 ...
%e A115154 466 = a(4,3) = a(4,2) + 3*a(3,3) = 121 + 3*115.
%Y A115154 Row sums give A115187.
%K A115154 nonn,easy,tabl
%O A115154 0,3
%A A115154 _Wolfdieter Lang_, Feb 23 2006