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 A335309 #18 Jan 09 2023 16:18:12
%S A335309 1,3,22,245,3606,65527,1411404,35066313,985483270,30869546411,
%T A335309 1065442493556,40144438269949,1638733865336764,72012798200637855,
%U A335309 3388250516614331416,169894851136173584145,9041936334960057699654,508945841697238471315027,30202327515992972576218980
%N A335309 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).
%H A335309 Seiichi Manyama, <a href="/A335309/b335309.txt">Table of n, a(n) for n = 0..381</a>
%F A335309 a(n) = central coefficient of (1 + (n + 2)*x + (n + 1)*x^2)^n.
%F A335309 a(n) = [x^n] 1 / sqrt(1 - 2*(n + 2)*x + n^2*x^2).
%F A335309 a(n) = n! * [x^n] exp((n + 2)*x) * BesselI(0,2*sqrt(n + 1)*x).
%F A335309 a(n) = Sum_{k=0..n} binomial(n,k)^2 * (n+1)^k.
%F A335309 a(n) ~ exp(2*sqrt(n)) * n^(n - 1/4) / (2*sqrt(Pi)) * (1 + 11/(12*sqrt(n))). - _Vaclav Kotesovec_, Jan 09 2023
%t A335309 Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] n^(n - k), {k, 0, n}], {n, 1, 18}]]
%t A335309 Table[SeriesCoefficient[1/Sqrt[1 - 2 (n + 2) x + n^2 x^2], {x, 0, n}], {n, 0, 18}]
%t A335309 Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 Sqrt[n + 1] x], {x, 0, n}], {n, 0, 18}]
%t A335309 Table[Hypergeometric2F1[-n, -n, 1, 1 + n], {n, 0, 18}]
%o A335309 (PARI) a(n) = sum(k=0, n, binomial(n,k)^2*(n+1)^k); \\ _Michel Marcus_, Jun 01 2020
%Y A335309 Cf. A001850, A069835, A084771, A084772, A098659, A187021, A307883, A331656, A335310.
%K A335309 nonn
%O A335309 0,2
%A A335309 _Ilya Gutkovskiy_, May 31 2020