A370441 Expansion of g.f. A(x) satisfying A(x) = A( x^3 + 3*A(x)^4 )^(1/3), with A(0)=0, A'(0)=1.

1, 1, 3, 12, 54, 261, 1324, 6952, 37461, 205977, 1151034, 6518085, 37321748, 215714904, 1256889150, 7374790400, 43537323406, 258417908640, 1541250594499, 9231988699115, 55514033703450, 334993491267955, 2027954403410504, 12312557796833622, 74955173794196890, 457431093085335708

Offset: 1

Author

Paul D. Hanna, Mar 09 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 + x^2 + 3*x^3 + 12*x^4 + 54*x^5 + 261*x^6 + 1324*x^7 + 6952*x^8 + 37461*x^9 + 205977*x^10 + 1151034*x^11 + 6518085*x^12 + ...
where A(x)^3 = A( x^3 + 3*A(x)^4 ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 55*x^6 + 270*x^7 + 1386*x^8 + 7347*x^9 + 39897*x^10 + 220779*x^11 + 1240392*x^12 + ...
A(x)^4 = x^4 + 4*x^5 + 18*x^6 + 88*x^7 + 451*x^8 + 2388*x^9 + 12958*x^10 + 71668*x^11 + 402489*x^12 + ...
Let B(x) denote the series reversion of A(x), A(B(x)) = x, where
B(x) = x - x^2 - x^3 - 2*x^4 - 4*x^5 - 9*x^6 - 22*x^7 - 55*x^8 - 142*x^9 - 376*x^10 - 1011*x^11 - 2758*x^12 + ... + (-1)^(n+1)*A107092(n)*x^n + ...
then B(x)^3 = B(x^3) - 3*x^4, where
B(x)^3 = x^3 - 3*x^4 - x^6 - x^9 - 2*x^12 - 4*x^15 - 9*x^18 - 22*x^21 - 55*x^24 - 142*x^27 - 376*x^30 - 1011*x^33 - 2758*x^36 + ...
Also, we have D(x) = x/B(x) is the g.f. of A091190, which begins
D(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 97*x^6 + 273*x^7 + 778*x^8 + 2240*x^9 + 6499*x^10 + 18976*x^11 + ... + A091190(n)*x^n + ...
such that D(x)^3 = D(x^3)/(1 - 3*x*D(x^3)).
		

Crossrefs

Cf. A107092, A091190, A370440.

Programs

PARI

{a(n) = my(A=[1],G); for(i=1,n, G = x*Ser(A); A = Vec((subst(G,x, x^3 + 3*x^2*G^2) + x^4*O(x^#A))^(1/3)); );A[n+1]}
for(n=0,40, print1(a(n),", "))

Formula

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

(1) A(x) = A( x^3 + 3*A(x)^4 )^(1/3).

(2) B(x)^3 = B(x^3) - 3*x^4, where A(B(x)) = x.

(3) A(x) = x*D(A(x)) where D(x) = x/Series_Reversion(A(x)) is the g.f. of A091190.