A276316 - OEIS

%I #30 Oct 15 2019 09:20:16

%S 1,4,31,300,3251,37744,459060,5773548,74474455,979872036,13099102575,

%T 177414673488,2429310288468,33574008073120,467717206216760,

%U 6560977611629676,92595131510426943,1313820730347196300,18730821529411507725,268185082351558093260

%N G.f. A(x) satisfies: x = A(x)-4*A(x)^2+A(x)^3.

%H Robert Israel, <a href="/A276316/b276316.txt">Table of n, a(n) for n = 1..844</a>

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

%H Thomas M. Richardson, <a href="http://arxiv.org/abs/1609.01193">The three 'R's and the Riordan dual</a>, arXiv:1609.01193 [math.CO], 2016.

%F G.f.: Series_Reversion(x-4*x^2+x^3).

%F From _Robert Israel_, Sep 02 2016: (Start)

%F G.f. g(x) satisfies the differential equation

%F (12-184*t-27*t^2)*g''(t) - (92+27*t)*g'(t) + 3*g(t) = 4.

%F (-27*n^2+3)*a(n)+(-184*n^2-276*n-92)*a(n+1)+(12*n^2+36*n+24)*a(n+2) = 0

%F for n >= 1. (End)

%F a(n) ~ (46 + 13*sqrt(13))^(n - 1/2) / (13^(1/4) * sqrt(Pi) * n^(3/2) * 2^(n + 1/2) * 3^(n - 1/2)). - _Vaclav Kotesovec_, Aug 22 2017

%e G.f.: A(x) = x+4*x^2+31*x^3+300*x^4+3251*x^5+37744*x^6+459060*x^7+...

%e Related Expansions:

%e A(x)^2 = x^2+8*x^3+78*x^4+848*x^5+9863*x^6+120096*x^7+1511634*x^8+...

%e A(x)^3 = x^3+12*x^4+141*x^5+1708*x^6+21324*x^7+272988*x^8+3566761*x^9+...

%p S:= series(RootOf(x-4*x^2+x^3-t,x),t,100):

%p seq(coeff(S,t,j),j=1..100); # _Robert Israel_, Sep 02 2016

%t Rest[CoefficientList[InverseSeries[Series[x - 4*x^2 + x^3, {x, 0, 20}], x],x]] (* _Vaclav Kotesovec_, Aug 22 2017 *)

%o (PARI) {a(n)=polcoeff(serreverse(x - 4*x^2 + x^3 + x^2*O(x^n)), n)}

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

%Y Cf. A250886, A276310, A276314, A276315.

%K nonn,easy

%O 1,2

%A _Tom Richardson_, Aug 29 2016