OFFSET
0,2
COMMENTS
Binomial transform of A005159.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..901
N. J. A. Sloane, Transforms
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(7*x) * (BesselI(0,6*x) - BesselI(1,6*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Catalan(k).
a(n) = 2F1([1/2, -n], [2], -12), where 2F1 is the hypergeometric function.
D-finite with recurrence (n+1) * a(n) = 2 * (7*n-3) * a(n-1) - 13 * (n-1) * a(n-2) for n > 1. - Seiichi Manyama, Jan 31 2021
a(n) ~ 13^(n + 3/2) / (8 * 3^(3/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 14 2021
MATHEMATICA
a[n_] := a[n] = 1 + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 20}]
Table[Hypergeometric2F1[1/2, -n, 2, -12], {n, 0, 20}]
PROG
(PARI) {a(n) = sum(k=0, n, 3^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))} \\ Seiichi Manyama, Jan 31 2021
(PARI) my(N=20, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)*(1-13*x)))) \\ Seiichi Manyama, Feb 01 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 28 2021
STATUS
approved