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%I #13 Sep 26 2023 04:14:01
%S 0,1,6,57,658,8442,115692,1658505,24565530,372999198,5774883348,
%T 90821581578,1446901409268,23301338376916,378711707274072,
%U 6203898306232233,102329366764727658,1698047225583890550,28327664136201303300,474821679792884860590,7992739387298462213340
%N G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 3 * A(x)).
%C Reversion of g.f. for hexagonal numbers (with signs).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SeriesReversion.html">Series Reversion</a>
%F 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 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 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 CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^3, {x, 0, 20}], x], x]
%t 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 Cf. A000384, A064087, A113207, A179848, A263843, A365817, A365818.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Sep 25 2023