A365816 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 3 * A(x)).

0, 1, 6, 57, 658, 8442, 115692, 1658505, 24565530, 372999198, 5774883348, 90821581578, 1446901409268, 23301338376916, 378711707274072, 6203898306232233, 102329366764727658, 1698047225583890550, 28327664136201303300, 474821679792884860590, 7992739387298462213340

Offset

0,3

Comments

Reversion of g.f. for hexagonal numbers (with signs).

Links

Table of n, a(n) for n=0..20.

Eric Weisstein's World of Mathematics, Hexagonal Number

Eric Weisstein's World of Mathematics, Series Reversion

Formula

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.

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

Mathematica

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]
CoefficientList[InverseSeries[Series[x (1 - 3 x)/(1 + x)^3, {x, 0, 20}], x], x]	
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}]]

Crossrefs

Cf. A000384, A064087, A113207, A179848, A263843, A365817, A365818.

Sequence in context: A130565 A124556 A369514 * A207412 A362167 A324447

Adjacent sequences: A365813 A365814 A365815 * A365817 A365818 A365819

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 25 2023

Status

approved