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 A345367 #39 Aug 05 2021 13:02:38
%S A345367 1,2,6,28,168,1137,8221,62041,482773,3845033,31188921,256757719,
%T A345367 2139691083,18015030073,153008796673,1309402039993,11279339531413,
%U A345367 97724562251137,851035285261745,7445189624293545,65401191955640665,576639234410182210,5101317352349364430
%N A345367 a(n) = Sum_{k=0..n} binomial(4*k,k) / (3*k + 1).
%C A345367 Partial sums of A002293.
%H A345367 Seiichi Manyama, <a href="/A345367/b345367.txt">Table of n, a(n) for n = 0..1000</a>
%F A345367 G.f. A(x) satisfies: A(x) = 1 / (1 - x) + x * (1 - x)^3 * A(x)^4.
%F A345367 a(n) ~ 2^(8*n + 17/2) / (229 * sqrt(Pi) * n^(3/2) * 3^(3*n + 3/2)). - _Vaclav Kotesovec_, Jul 28 2021
%F A345367 D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) +(-283*n^3+384*n^2-173*n+24)*a(n-1) +8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-2)=0. - _R. J. Mathar_, Aug 05 2021
%t A345367 Table[Sum[Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 22}]
%t A345367 nmax = 22; A[_] = 0; Do[A[x_] = 1/(1 - x) + x (1 - x)^3 A[x]^4 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%o A345367 (PARI) a(n) = sum(k=0, n, binomial(4*k, k)/(3*k+1)); \\ _Michel Marcus_, Jul 28 2021
%Y A345367 Cf. A002293, A014137, A104859, A345368, A346065, A346646, A346671, A346672.
%K A345367 nonn
%O A345367 0,2
%A A345367 _Ilya Gutkovskiy_, Jul 28 2021