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 A257633 #16 Apr 11 2025 08:45:54
%S A257633 1,6,45,364,3060,26334,230230,2035800,18156204,163011640,1471442973,
%T A257633 13340783196,121399651100,1108176102180,10142940735900,93052749919920,
%U A257633 855420636763836,7877932561061640,72667580816130436,671262558647881200,6208770443303347920
%N A257633 a(n) = binomial(4*n + 2,n).
%H A257633 Michael De Vlieger, <a href="/A257633/b257633.txt">Table of n, a(n) for n = 0..1025</a>
%H A257633 N. J. Wildberger and Dean Rubine, <a href="https://doi.org/10.1080/00029890.2025.2460966">A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode</a>, Amer. Math. Monthly (2025). See section 12.
%F A257633 The o.g.f. equals f(x)*g(x)^2, where f(x) is the o.g.f. for A005810 and g(x) is the o.g.f. for A002293. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(4*n + k,n). Cf. A262977 (k = -1), A005810 (k = 0), A052203 (k = 1), A224274 (k = 3) and A004331 (k = 4).
%p A257633 #A257633
%p A257633 seq(binomial(4*n + 2,n), n = 0..20);
%t A257633 Table[Binomial[4*n + 2, n], {n, 0, 120}] (* _Michael De Vlieger_, Apr 11 2025 *)
%o A257633 (PARI) vector(30, n, n--; binomial(4*n+2, n)) \\ _Altug Alkan_, Nov 05 2015
%Y A257633 Cf. A002293, A004331, A005810, A052203, A224274, A262977.
%K A257633 nonn,easy
%O A257633 0,2
%A A257633 _Peter Bala_, Nov 04 2015