A370439 Expansion of g.f. A(x) satisfying A(x) = A( x*A(x)^2 + 3*x*A(x)^3 )^(1/3).

1, 3, 9, 30, 126, 648, 3591, 19953, 110079, 610500, 3440493, 19742616, 114918138, 675417474, 3996992547, 23791052862, 142393544757, 856746349992, 5179722791274, 31449875426622, 191678795532801, 1172198278949454, 7190652243631437, 44235165115911312, 272837082264574914

Offset: 1

Author

Paul D. Hanna, Mar 27 2024

Keywords

• nonn

Comments

Compare the g.f. to the following identities:

(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),

(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),

where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + 3*x^2 + 9*x^3 + 30*x^4 + 126*x^5 + 648*x^6 + 3591*x^7 + 19953*x^8 + 110079*x^9 + 610500*x^10 + 3440493*x^11 + 19742616*x^12 + ...
where A(x)^3 = A( x*A(x)^2 + 3*x*A(x)^3 ).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A356782, A117940, A370546.

Programs

PARI

{a(n) = my(A=[0,1]); for(i=1,n, A=concat(A,0);
F=Ser(A); A[#A] = polcoeff(subst(F,x,x*F^2 + 3*x*F^3) - F^3,#A+1) );A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1.a) A(x)^3 = A( x*A(x)^2 * (1 + 3*A(x)) ).

(1.b) A(x)^9 = A( x*A(x)^8 * (1 + 3*A(x))*(1 + 3*A(x)^3) ).

(1.c) A(x)^27 = A( x*A(x)^26 * (1 + 3*A(x))*(1 + 3*A(x)^3)*(1 + 3*A(x)^9) ).

(1.d) A(x)^(3^n) = A( x*A(x)^(3^n-1) * Product_{k=0..n-1} (1 + 3*A(x)^(3^k)) ).

(2) A(x) = x * Product_{n>=0} (1 + 3*A(x)^(3^n)).

(3) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 3*x^(3^n)) ).

(4) A(x) = x * Sum_{n>=0} A117940(n) * A(x)^n, where g.f. of A117940 equals Product{k>=0} 1 + 3*x^(3^k).

a(n) ~ c * d^n / n^(3/2), where d = 6.5583689184153129045048... and c = 0.129061736750222730297... - Vaclav Kotesovec, Apr 05 2024

The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3^(n+1) * A(r)^(3^n) / (1 + 3*A(r)^(3^n)) and r = A(r) / Product_{n>=0} (1 + 3*A(r)^(3^n)), where r = 0.1524769363297159918479... = 1/d (d is given above) and A(r) = 0.3905308673397427979651361312666180120359942797557... - Paul D. Hanna, May 22 2024