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 A064340 #17 Apr 21 2025 02:41:25
%S A064340 1,1,4,28,256,2704,31168,380608,4840960,63458560,851399680,
%T A064340 11635096576,161396604928,2266669453312,32166082822144,
%U A064340 460531091685376,6644185553305600,96498260064403456,1409750653282287616,20702370737659052032,305428492830594039808
%N A064340 Generalized Catalan numbers C(2,2; n).
%C A064340 See triangle A064879 with columns m built from C(m,m; n), m >= 0, also for Derrida et al. and Liggett references.
%H A064340 J. Abate and W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Whitt/whitt6.html">Brownian Motion and the Generalized Catalan Numbers</a>, J. Int. Seq. 14 (2011) # 11.2.6, corollary 6.
%F A064340 a(n) = ((4^(n-1))/(n-1))*Sum_{m=0..n-2} (m+1)*(m+2)*binomial(2*(n-2)-m, n-2-m)/2^(m+1), n >= 2, a(0) = a(1) = 1.
%F A064340 G.f.: (1-3*x*c(4*x))/(1-2*x*c(4*x))^2 = c(4*x)*(3+c(4*x))/(1+c(4*x))^2 = (1+5*x+3*c(4*x)*(2*x)^2)/(1+2*x)^2 with c(x) = A(x) g.f. of Catalan numbers A000108.
%F A064340 (-n+1)*a(n) + 2*(7*n-20)*a(n-1) + 16*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Aug 09 2017
%o A064340 (PARI) my(x='x+O('x^30)); Vec((1+(13-3*sqrt(1-16*x))*x/2)/(1+2*x)^2) \\ _Jinyuan Wang_, Apr 20 2025
%Y A064340 Cf. A000108 (Catalan as C(1,1; n)), A064879, A067298.
%K A064340 nonn,easy
%O A064340 0,3
%A A064340 _Wolfdieter Lang_, Oct 12 2001