A374567 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).

1, 2, 9, 51, 325, 2222, 15926, 118085, 898217, 6970053, 54960439, 439112322, 3547096393, 28921270773, 237704587991, 1967321998468, 16381661824340, 137144132047520, 1153655788549216, 9746264972136632, 82656795697147384, 703459159019830315, 6005956718852682504, 51426768620398474939

Offset: 1

Author

Paul D. Hanna, Aug 13 2024

Keywords

• nonn

Comments

Compare to: C(x)^2 = C( x^2 - 2*C(x)^3 ), where C(x) = x - C(x)^2.

Examples

			G.f.: A(x) = x + 2*x^2 + 9*x^3 + 51*x^4 + 325*x^5 + 2222*x^6 + 15926*x^7 + 118085*x^8 + 898217*x^9 + 6970053*x^10 + ...
where A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).
RELATED SERIES.
Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then
A( x - x^2 - x*G(x) ) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) begins
G(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + 207*x^11 + 451*x^12 + 983*x^13 + ... + A001190(n)*x^n + ...
A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 138*x^5 + 935*x^6 + 6662*x^7 + 49191*x^8 + 373020*x^9 + 2887711*x^10 + 22727256*x^11 + ...
A(x)^3 = x^3 + 6*x^4 + 39*x^5 + 269*x^6 + 1938*x^7 + 14418*x^8 + 109932*x^9 + 854568*x^10 + 6747672*x^11 + ...
SPECIFIC VALUES.
A(t) = 1/5 at t = 0.1094430388151747748055350980742058560407673560783455...
where 1/25 = A( t^2 + 2*t/25 + 2/125 ).
A(t) = 1/6 at t = 0.1053569291935061227625330002451062383852684202941979...
where 1/36 = A( t^2 + t/18 + 1/108 ).
A(1/10) = 0.1471263013840628871589336795118257882025452972700045...
where A(1/10)^2 = A( 1/10^2 + (2/10)*A(1/10)^2 + 2*A(1/10)^3 ).
A(1/11) = 0.1237258078822115859596611191115342221543387518134407...
A(1/12) = 0.1081759735424269717469930892718654709953905803313352...
		

Links

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

Crossrefs

Cf. A374566, A271959, A001190.

Programs

PARI

{a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2 + 2*x*Ax^2 + 2*Ax^3) - Ax^2, #A) ); 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(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^3 ).

(2) x = A( x - x^2 - x*G(x) ), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of A001190, the Wedderburn-Etherington numbers.

(3) x^2 = A( x^2*(1 - G(x))^2 + 2*x^3 - x^4 ), where G(x) is the g.f. of A001190.

(4) x = A( x*sqrt(1 - 2*x - G(x^2)) - x^2 ), where G(x) is the g.f. of A001190.