A120590 G.f. satisfies: 4*A(x) = 3 + x + A(x)^3, starting with [1,1,3].

1, 1, 3, 19, 150, 1326, 12558, 124590, 1278189, 13449205, 144342627, 1573990275, 17389407984, 194228357568, 2189610888840, 24881753664840, 284708154606318, 3277578288381318, 37934510719585350, 441152315040444150

Offset: 0

Author

Paul D. Hanna, Jun 16 2006, Jan 24 2008

Keywords

• nonn

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Examples

			A(x) = 1 + x + 3*x^2 + 19*x^3 + 150*x^4 + 1326*x^5 + 12558*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 76*x^3 + 600*x^4 + 5304*x^5 + 50232*x^6 +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..70
• Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.

Crossrefs

Cf. A120591 (A(x)^3); A244594, A120588, A120592 - A120607.

Programs

Mathematica

FullSimplify[Table[SeriesCoefficient[Sum[Binomial[3*k,k]/(2*k+1)*(3+x)^(2*k+1)/4^(3*k+1),{k,0,Infinity}],{x,0,n}] ,{n,0,20}]] (* Vaclav Kotesovec, Oct 19 2012 *)

PARI

{a(n)=local(A=1+x+3*x^2+x*O(x^n));for(i=0,n,A=A-4*A+3+x+A^3);polcoeff(A,n)}

Formula

G.f.: A(x) = 1 + Series_Reversion(1+4*x - (1+x)^3).

G.f.: A(x) = Sum_{n>=0} C(3*n,n)/(2*n+1) * (3+x)^(2*n+1) / 4^(3*n+1), due to Lagrange Inversion.

Recurrence: 13*(n-1)*n*a(n) = 81*(n-1)*(2*n-3)*a(n-1) + 3*(3*n-7)*(3*n-5)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ sqrt(32-18*sqrt(3))*((81+48*sqrt(3))/13)^n/(12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

G.f.: 4 * sin( arcsin(3 * sqrt(3) * (3 + x) / 16) / 3) / sqrt(3). - Benedict W. J. Irwin, Oct 19 2016