OFFSET
0,2
COMMENTS
Inverse binomial transform of A005159.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..964
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
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 28 2021
STATUS
approved