A250887 G.f. A(x) satisfies: x = A(x) * (1 + A(x)) * (1 - 3*A(x)).

1, 2, 11, 70, 503, 3864, 31092, 258654, 2206655, 19200610, 169739843, 1520241320, 13764959908, 125792608400, 1158745944312, 10747830197070, 100295912869263, 940958196049830, 8870071185895425, 83972749650989430, 798033019890224415, 7610570090722324320, 72810031747355657040

Offset: 1

Author

Paul D. Hanna, Nov 28 2014

Keywords

• nonn

Examples

			G.f.: A(x) = x + 2*x^2 + 11*x^3 + 70*x^4 + 503*x^5 + 3864*x^6 + ...
Related expansions.
A(x)^2 = x^2 + 4*x^3 + 26*x^4 + 184*x^5 + 1407*x^6 + 11280*x^7 + ...
A(x)^3 = x^3 + 6*x^4 + 45*x^5 + 350*x^6 + 2844*x^7 + 23814*x^8 + ...
where x = A(x) - 2*A(x)^2 - 3*A(x)^3.
The square-root of A(x)/x is the g.f. of A222050:
sqrt(A(x)/x) = 1 + x + 5*x^2 + 30*x^3 + 209*x^4 + 1573*x^5 + 12478*x^6 + ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..995
• Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
• Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.

Crossrefs

Cf. A222050.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - 2*x^2 - 3*x^3, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Aug 21 2017 *)

PARI

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

Formula

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

a(n) ~ (13*sqrt(13) + 35)^(n-1/2) / (13^(1/4) * sqrt(Pi) * n^(3/2) * 2^(3*n-1/2)). - Vaclav Kotesovec, Aug 21 2017