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 A113647 #15 Aug 06 2024 09:00:43
%S A113647 1,1,3,1,7,13,1,15,41,67,1,31,113,247,381,1,63,289,783,1545,2307,1,
%T A113647 127,705,2271,5361,9975,14589,1,255,1665,6207,16929,36879,66057,95235,
%U A113647 1,511,3841,16255,50113,123871,255985,446455,636925,1,1023,8705,41215,141441
%N A113647 Triangle of numbers related to the generalized Catalan sequence C(2;n+1)=A064062(n+1), n>=0.
%C A113647 This triangle, called Y(2,1), appears in the totally asymmetric exclusion process for the (unphysical) values alpha=2, 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 A113647 The main diagonal (M=1) gives the generalized Catalan sequence C(2,n):=A064062(n).
%C A113647 The diagonal sequences give A064062(n+1), A115137, A115150-A115153, for n+1>= M=1,..,6.
%H A113647 B. Derrida, E. Domany and D. Mukamel, <a href="https://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 A113647 Wolfdieter Lang, <a href="/A113647/a113647.txt">First 10 rows</a>.
%F A113647 a(n, n+1)=A064062(n+1) (main diagonal with M=1); a(n, n-M+2)= a(n, n-M+1) + 2*a(n-1, n-M+2), M>=2; a(n, 1)=1; n>=0.
%F A113647 G.f. for diagonal sequence M=1: GY(1, x):=(2*c(2*x)-1)/(1+x) with c(x) g.f. of A000108 (Catalan); for M=2: GY(2, x)=(1-2*x)*GY(1, x)-1; for M>=3: GY(M, x)= GY(M-1, x) -2*x*GY(M-2, x) + x^(M-2).
%F A113647 G.f. for diagonal sequence M (solution to the above given recurrence): GY(M, x)= (x^(M-1)/(1+x))*( 2^(M+1)*x*(p(M, 2*x)-(2*x)*p(M+1, 2*x)*c(2*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 A113647 Triangle begins:
%e A113647 1;
%e A113647 1,3;
%e A113647 1,7,13;
%e A113647 1,15,41,67;
%e A113647 1,31,113,247,381;
%e A113647 ...
%e A113647 113=a(4,3)= a(4,2) + 2*a(3,3)= 31 + 2*41.
%Y A113647 Row sums give A115136.
%K A113647 nonn,easy,tabl
%O A113647 0,3
%A A113647 _Wolfdieter Lang_, Jan 13 2006