OFFSET
1,2
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..955
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
FORMULA
G.f.: Series_Reversion(x - 3*x^2 + x^3).
G.f. A(x) satisfies:
(1) 1/x = Sum_{n>=1} Fibonacci(2*n) * A(x)^(n-2).
(2) 1+x = 2*(1-A(x)) - (1-A(x))^3.
(3) 5+x = 10*(1+A(x)) - 6*(1+A(x))^2 + (1+A(x))^3.
a(n) = Sum_{k=0..n-1} binomial(n+k-1,k)*binomial(3*n+k-2,n-k-1))/n. - Vladimir Kruchinin, Mar 11 2015
a(n) = binomial(3*n-2,n-1)*hypergeom([1-n,3*n-1],[n+1/2],-1/4)/n. - Peter Luschny, Mar 11 2015
5*n*(n-1)*a(n) - 27*(n-1)*(2*n-3)*a(n-1) - 3*(3*n-5)*(3*n-7)*a(n-2) = 0. - R. J. Mathar, Jul 15 2017
a(n) ~ 3^(n - 3/4) * (9 + 4*sqrt(6))^(n - 1/2) / (2^(5/4) * sqrt(Pi) * n^(3/2) * 5^(n - 1/2)). - Vaclav Kotesovec, Aug 22 2017
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 17*x^3 + 120*x^4 + 948*x^5 + 8022*x^6 + ...
Related expansions.
A(x)^2 = x^2 + 6*x^3 + 43*x^4 + 342*x^5 + 2905*x^6 + 25812*x^7 + ...
A(x)^3 = x^3 + 9*x^4 + 78*x^5 + 693*x^6 + 6330*x^7 + 59211*x^8 + ...
where x = A(x) - 3*A(x)^2 + A(x)^3.
Also, the g.f. satisfies:
1/x = 1/A(x) + 3 + 8*A(x) + 21*A(x)^2 + 55*A(x)^3 + 144*A(x)^4 + 377*A(x)^5 + 987*A(x)^6 + ... + Fibonacci(2*n) * A(x)^(n-2) + ...
MAPLE
a := n -> binomial(3*n-2, n-1)*hypergeom([1-n, 3*n-1], [n+1/2], -1/4)/n:
seq(simplify(a(n)), n=1..23); # Peter Luschny, Mar 11 2015
MATHEMATICA
a[n_] := Sum[Binomial[n+k-1, k]*Binomial[3*n+k-2, n-k-1], {k, 0, n-1}]/n; Array[a, 30] (* Jean-François Alcover, Mar 11 2015, after Vladimir Kruchinin *)
Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 + x^3, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Aug 22 2017 *)
PROG
(PARI) {a(n)=polcoeff(serreverse(x - 3*x^2 + x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))
(Maxima)
a(n):=sum(binomial(n+k-1, k)*binomial(3*n+k-2, n-k-1), k, 0, n-1)/n; /* Vladimir Kruchinin, Mar 11 2015 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 28 2014
STATUS
approved