A374566 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2 + 2*(1+x)*A(x)^3 ).

1, 1, 4, 16, 76, 381, 2010, 10955, 61265, 349472, 2025632, 11896039, 70632739, 423300099, 2557174039, 15555534859, 95202925651, 585799778042, 3621806301246, 22488577587970, 140176525844646, 876813040040057, 5501997007343589, 34625517090342459, 218489435424317825, 1382072993052136903

Offset: 1

Author

Paul D. Hanna, Aug 12 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 + x^2 + 4*x^3 + 16*x^4 + 76*x^5 + 381*x^6 + 2010*x^7 + 10955*x^8 + 61265*x^9 + 349472*x^10 + ...
where A(x)^2 = A( x^2 + 2*(1+x)*A(x)^3 ).
RELATED SERIES.
Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then
A( x - x^3 - 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 + 2*x^3 + 9*x^4 + 40*x^5 + 200*x^6 + 1042*x^7 + 5646*x^8 + 31410*x^9 + 178488*x^10 + 1031346*x^11 + 6041569*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 15*x^5 + 73*x^6 + 384*x^7 + 2079*x^8 + 11584*x^9 + 65868*x^10 + 380859*x^11 + 2232199*x^12 + 13231686*x^13 + ...
x^2 + 2*(1+x)*A(x)^3 = x^2 + 2*x^3 + 8*x^4 + 36*x^5 + 176*x^6 + 914*x^7 + 4926*x^8 + 27326*x^9 + 154904*x^10 + 893454*x^11 + ...
SPECIFIC VALUES.
A(t) = 1/4 at t = 0.14894182268166520428651100246692394784806895864208130...
where 1/16 = A( t^2 + (1 + t)/32 ).
A(t) = 1/5 at t = 0.14144303881517477480553509807420585604076735607834555...
where 1/25 = A( t^2 + 2*(1 + t)/125 ).
A(1/7) = 0.204913420188897006601259679664181034021504614738141...
where A(1/7)^2 = A( 1/7^2 + (16/7)*A(1/7)^3 ).
A(1/8) = 0.159462997675623738517233384699423553894402512640906...
where A(1/8)^2 = A( 1/8^2 + (18/8)*A(1/8)^3 ).
A(1/9) = 0.134511672187656270338825814076702307725993232871545...
where A(1/9)^2 = A( 1/9^2 + (20/9)*A(1/9)^3 ).
A(1/10) = 0.117197825788422212715965141990212003609448403429416...
where A(1/10)^2 = A( 1/10^2 + (22/10)*A(1/10)^3 ).
		

Links

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

Crossrefs

Cf. A374567, 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*(1+x)*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*(1+x)*A(x)^3 ).

(2) x = A( x - x^3 - 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^6 ), where G(x) is the g.f. of A001190.

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