A249924 G.f. A(x) satisfies: x = A(x) - 3*A(x)^2 + A(x)^3.

1, 3, 17, 120, 948, 8022, 71106, 651717, 6126175, 58736535, 572178165, 5647102500, 56345894052, 567444190680, 5760259701864, 58879552102416, 605508278430348, 6260413541738610, 65036607553643550, 678530086525374930, 7106457298203380370, 74688269331406258260, 787463653336202248380

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

Sequence in context: A344553 A121572 A340993 * A305307 A074543 A216314

Adjacent sequences: A249921 A249922 A249923 * A249925 A249926 A249927

Keyword

nonn

Author

Paul D. Hanna, Nov 28 2014

Status

approved