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 A365816 #14 Feb 16 2025 08:34:06
%S A365816 0,1,6,57,658,8442,115692,1658505,24565530,372999198,5774883348,
%T A365816 90821581578,1446901409268,23301338376916,378711707274072,
%U A365816 6203898306232233,102329366764727658,1698047225583890550,28327664136201303300,474821679792884860590,7992739387298462213340
%N A365816 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 3 * A(x)).
%C A365816 Reversion of g.f. for hexagonal numbers (with signs).
%H A365816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>
%H A365816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SeriesReversion.html">Series Reversion</a>
%F A365816 a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3*n,n-k-1) * 3^k for n > 0.
%F A365816 a(n) ~ 6^(3*n + 1/2) / (sqrt((481 + 133*sqrt(13))*Pi) * n^(3/2) * (13*sqrt(13) - 35)^n). - _Vaclav Kotesovec_, Sep 26 2023
%t A365816 nmax = 20; A[_] = 0; Do[A[x_] = x (1 + A[x])^3/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A365816 CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^3, {x, 0, 20}], x], x]
%t A365816 Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 n, n - k - 1] 3^k, {k, 0, n - 1}], {n, 1, 20}]]
%Y A365816 Cf. A000384, A064087, A113207, A179848, A263843, A365817, A365818.
%K A365816 nonn
%O A365816 0,3
%A A365816 _Ilya Gutkovskiy_, Sep 25 2023