A276316 G.f. A(x) satisfies: x = A(x)-4*A(x)^2+A(x)^3.

1, 4, 31, 300, 3251, 37744, 459060, 5773548, 74474455, 979872036, 13099102575, 177414673488, 2429310288468, 33574008073120, 467717206216760, 6560977611629676, 92595131510426943, 1313820730347196300, 18730821529411507725, 268185082351558093260

Offset: 1

Author

Tom Richardson, Aug 29 2016

Keywords

• nonn
• easy

Examples

			G.f.: A(x) = x+4*x^2+31*x^3+300*x^4+3251*x^5+37744*x^6+459060*x^7+...
Related Expansions:
A(x)^2 = x^2+8*x^3+78*x^4+848*x^5+9863*x^6+120096*x^7+1511634*x^8+...
A(x)^3 = x^3+12*x^4+141*x^5+1708*x^6+21324*x^7+272988*x^8+3566761*x^9+...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..844
• 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.

Crossrefs

Cf. A250886, A276310, A276314, A276315.

Programs

Maple

S:= series(RootOf(x-4*x^2+x^3-t,x),t,100):
seq(coeff(S,t,j),j=1..100); # Robert Israel, Sep 02 2016

Mathematica

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

PARI

{a(n)=polcoeff(serreverse(x - 4*x^2 + x^3 + x^2*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

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

From Robert Israel, Sep 02 2016: (Start)

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

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

(-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

for n >= 1. (End)

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