A250886 G.f. A(x) satisfies: x = A(x) * (1 + A(x)) * (1 - 2*A(x)).

1, 1, 4, 15, 68, 322, 1608, 8283, 43780, 235950, 1291992, 7167030, 40192488, 227488900, 1297845008, 7455558675, 43088726148, 250362137590, 1461641062200, 8569690323810, 50438119336440, 297896152159260, 1765010252344560, 10487875429825950, 62485899131628648, 373198022044163532

Offset: 1

Author

Paul D. Hanna, Nov 28 2014

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 15*x^4 + 68*x^5 + 322*x^6 + 1608*x^7 + ...
Related expansions.
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 38*x^5 + 182*x^6 + 900*x^7 + 4629*x^8 + ...
A(x)^3 = x^3 + 3*x^4 + 15*x^5 + 70*x^6 + 354*x^7 + 1827*x^8 + 9691*x^9 + ...
where x = A(x) - A(x)^2 - 2*A(x)^3.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1254
• Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
• Thomas M. Richardson, The three 'R's and the Riordan dual, arXiv:1609.01193 [math.CO], 2016.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - x^2 - 2*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)

PARI

{a(n)=polcoeff(serreverse(x - x^2 - 2*x^3 + x^2*O(x^n)), n)}
for(n=1,30,print1(a(n),", "))

Formula

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

G.f. A(x) satisfies: x = -3*(1+A(x)) + 5*(1+A(x))^2 - 2*(1+A(x))^3.

a(n) ~ 2^(n - 3/2) * (10 + 7*sqrt(7))^(n - 1/2) / (7^(1/4) * sqrt(Pi) * n^(3/2) * 3^(2*n - 1)). - Vaclav Kotesovec, Aug 22 2017