A249924 - OEIS

%I #41 Nov 12 2024 09:03:50

%S 1,3,17,120,948,8022,71106,651717,6126175,58736535,572178165,

%T 5647102500,56345894052,567444190680,5760259701864,58879552102416,

%U 605508278430348,6260413541738610,65036607553643550,678530086525374930,7106457298203380370,74688269331406258260,787463653336202248380

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

%H Michael De Vlieger, <a href="/A249924/b249924.txt">Table of n, a(n) for n = 1..955</a>

%H Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.

%F G.f.: Series_Reversion(x - 3*x^2 + x^3).

%F G.f. A(x) satisfies:

%F (1) 1/x = Sum_{n>=1} Fibonacci(2*n) * A(x)^(n-2).

%F (2) 1+x = 2*(1-A(x)) - (1-A(x))^3.

%F (3) 5+x = 10*(1+A(x)) - 6*(1+A(x))^2 + (1+A(x))^3.

%F 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

%F 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

%F 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

%F 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

%e G.f.: A(x) = x + 3*x^2 + 17*x^3 + 120*x^4 + 948*x^5 + 8022*x^6 + ...

%e Related expansions.

%e A(x)^2 = x^2 + 6*x^3 + 43*x^4 + 342*x^5 + 2905*x^6 + 25812*x^7 + ...

%e A(x)^3 = x^3 + 9*x^4 + 78*x^5 + 693*x^6 + 6330*x^7 + 59211*x^8 + ...

%e where x = A(x) - 3*A(x)^2 + A(x)^3.

%e Also, the g.f. satisfies:

%e 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) + ...

%p a := n -> binomial(3*n-2,n-1)*hypergeom([1-n,3*n-1], [n+1/2],-1/4)/n:

%p seq(simplify(a(n)), n=1..23); #  _Peter Luschny_, Mar 11 2015

%t 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_ *)

%t Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 + x^3, {x, 0, 20}], x],x]] (* _Vaclav Kotesovec_, Aug 22 2017 *)

%o (PARI) {a(n)=polcoeff(serreverse(x - 3*x^2 + x^3 + x^2*O(x^n)), n)}

%o for(n=1, 30, print1(a(n), ", "))

%o (Maxima)

%o 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 */

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 28 2014