A337169 a(n) = (-1)^n + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).

1, 2, 13, 89, 691, 5720, 49555, 443630, 4071595, 38105342, 362271823, 3488988101, 33967656469, 333752559392, 3305347855573, 32960499084305, 330664662067795, 3335002912108670, 33796042027030855, 343940115478559699, 3513702627928096681, 36021007341027948032

Offset: 0

Author

Ilya Gutkovskiy, Jan 28 2021

Keywords

• nonn

Comments

Inverse binomial transform of A005159.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..964

Crossrefs

Cf. A000108, A005043, A005159, A337167, A337168.

Programs

Mathematica

a[n_] := a[n] = (-1)^n + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 21}]
Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 12], {n, 0, 21}]

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + 3*x*A(x)^2.

G.f.: (1 - sqrt(1 - 12*x / (1 + x))) / (6*x).

E.g.f.: exp(5*x) * (BesselI(0,6*x) - BesselI(1,6*x)).

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * Catalan(k).

a(n) ~ 11^(n + 3/2) / (8 * 3^(3/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021