A250888 - OEIS

%I #12 Oct 15 2019 12:21:35

%S 1,3,22,195,1938,20622,229836,2648547,31301050,377301210,4620769140,

%T 57333249870,719179311732,9105192433980,116197502184984,

%U 1493159297251491,19303993468386378,250907887026047010,3276818401723155300,42977976005402922330,565863442299520006620

%N G.f. A(x) satisfies: x = A(x) * (1 + A(x)) * (1 - 4*A(x)).

%H Michael De Vlieger, <a href="/A250888/b250888.txt">Table of n, a(n) for n = 1..873</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.

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

%F a(n) ~ 2^(n - 3/2) * 3^(n - 3/4) * (27 + 7*sqrt(21))^(n - 1/2) / (7^(1/4) * sqrt(Pi) * n^(3/2) * 5^(2*n - 1)). - _Vaclav Kotesovec_, Aug 22 2017

%e G.f.: A(x) = x + 3*x^2 + 22*x^3 + 195*x^4 + 1938*x^5 + 20622*x^6 +...

%e Related expansions.

%e A(x)^2 = x^2 + 6*x^3 + 53*x^4 + 522*x^5 + 5530*x^6 + 61452*x^7 +...

%e A(x)^3 = x^3 + 9*x^4 + 93*x^5 + 1008*x^6 + 11370*x^7 + 132111*x^8 +...

%e where x = A(x) - 3*A(x)^2 - 4*A(x)^3.

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

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

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

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 28 2014