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 A337168 #19 Jan 21 2025 09:24:01
%S A337168 1,1,5,21,105,553,3053,17405,101713,606033,3667797,22485477,139340985,
%T A337168 871429497,5492959293,34862161869,222592918689,1428814897825,
%U A337168 9215016141989,59684122637237,388045493943049,2531696701375689,16569559364596365,108758426952823709
%N A337168 a(n) = (-1)^n + 2 * Sum_{k=0..n-1} a(k) * a(n-k-1).
%C A337168 Inverse binomial transform of A151374.
%H A337168 Seiichi Manyama, <a href="/A337168/b337168.txt">Table of n, a(n) for n = 0..1000</a>
%H A337168 Paveł Szabłowski, <a href="https://cdm.ucalgary.ca/article/view/76214">Beta distributions whose moment sequences are related to integer sequences listed in the OEIS</a>, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See pp. 97-98.
%F A337168 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + 2*x*A(x)^2.
%F A337168 G.f.: (1 - sqrt(1 - 8*x / (1 + x))) / (4*x).
%F A337168 E.g.f.: exp(3*x) * (BesselI(0,4*x) - BesselI(1,4*x)).
%F A337168 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * Catalan(k).
%F A337168 a(n) ~ 7^(n + 3/2) / (2^(9/2) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Nov 13 2021
%t A337168 a[n_] := a[n] = (-1)^n + 2 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]
%t A337168 Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k CatalanNumber[k], {k, 0, n}], {n, 0, 23}]
%t A337168 Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 8], {n, 0, 23}]
%Y A337168 Cf. A000108, A005043, A052701, A151374, A162326, A337169.
%K A337168 nonn
%O A337168 0,3
%A A337168 _Ilya Gutkovskiy_, Jan 28 2021