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 A349513 #9 Nov 20 2021 20:58:02
%S A349513 1,3,12,56,294,1722,11256,82224,670662,6084578,61030536,672041328,
%T A349513 8067200092,104884001796,1468416141744,22026397243680,352422956979270,
%U A349513 5991192602253810,107841475915703880,2048988077743637520,40979761692719279220,860574996085362738060
%N A349513 a(n) = n! * Sum_{k=0..n} (2*k)! / (k!)^3.
%F A349513 E.g.f.: exp(2*x) * BesselI(0,2*x) / (1 - x).
%F A349513 a(n) = Sum_{k=0..n} binomial(n,k) * A000984(k) * A000142(n-k).
%F A349513 a(n) = Sum_{k=0..n} binomial(n,k) * A002426(k) * A000522(n-k).
%F A349513 a(n) ~ exp(2) * BesselI(0,2) * n!. - _Vaclav Kotesovec_, Nov 20 2021
%t A349513 Table[n! Sum[(2 k)!/(k!)^3, {k, 0, n}], {n, 0, 21}]
%t A349513 nmax = 21; CoefficientList[Series[Exp[2 x] BesselI[0, 2 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
%o A349513 (PARI) a(n) = n! * sum(k=0, n, (2*k)! / (k!)^3) \\ _Andrew Howroyd_, Nov 20 2021
%Y A349513 Cf. A000142, A000522, A000984, A002426, A026375, A052143, A336293.
%K A349513 nonn
%O A349513 0,2
%A A349513 _Ilya Gutkovskiy_, Nov 20 2021