A374572 Expansion of g.f. A(x) satisfying A(x)^2 = A( x*(1+x)*A(x) ).

1, 1, 1, 3, 5, 11, 27, 69, 183, 481, 1283, 3453, 9361, 25651, 70927, 197721, 555039, 1567345, 4449023, 12686465, 36323203, 104381397, 300958959, 870378337, 2524129349, 7338679127, 21386456807, 62459196233, 182776933033, 535861013939, 1573742036447, 4629306941913

Offset: 1

Author

Paul D. Hanna, Jul 18 2024

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 5*x^5 + 11*x^6 + 27*x^7 + 69*x^8 + 183*x^9 + 481*x^10 + 1283*x^11 + 3453*x^12 + 9361*x^13 + 25651*x^14 + 70927*x^15 + ...
where A(x)^2 = A( x*(1+x)*A(x) ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 17*x^6 + 38*x^7 + 95*x^8 + 244*x^9 + 649*x^10 + 1738*x^11 + 4699*x^12 + ...
Let B(x) be the series reversion of g.f. A(x), B(A(x)) = x, then
B(x) = x - x^2 + x^3 - 3*x^4 + 9*x^5 - 25*x^6 + 71*x^7 - 219*x^8 + 693*x^9 - 2197*x^10 + 7069*x^11 - 23135*x^12 + ...
where B(x^2) = x*B(x)*(1 + B(x)).
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.301949314609828865985839329094529550482897401344979...
where 1/4 = A( t*(1 + t)/2 ).
A(3/10) = 0.492388112365452715229250795508017422919418907801551...
where A(3/10)^2 = A( (39/100)*A(3/10) ).
A(2/7) = 0.443877424659041232765055763766392304444609934055603...
where A(2/7)^2 = A( (18/49)*A(2/7) ).
A(1/4) = 0.352241294433584221893793757577235288109595399125986...
where A(1/4)^2 = A( (5/16)*A(1/4) ).
A(1/5) = 0.255826785620580342641277164817159026900345909888978...
where A(1/5)^2 = A( (6/25)*A(1/5) ).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..520

Crossrefs

Cf. A075864.

Programs

PARI

{a(n) = my(A=[0,1], Ax); for(i=1,n, A = concat(A,0); Ax = Ser(A);
A[#A] = polcoeff( subst(Ax,x, x*(1+x)*Ax ) - Ax^2, #A) );A[n+1]}
for(n=1,40,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*(1+x)*A(x) ).

(2) A(x)^4 = A( x*(1+x)*A(x)^3 + x^2*(1+x)^2*A(x)^4 ).

(3) A(x)^8 = A( x*(1+x)*A(x)^7 + x^2*(1+x)^2*A(x)^8 + x^2*(1+x)^2*A(x)^10 + 2*x^3*(1+x)^3*A(x)^11 + x^4*(1+x)^4*A(x)^12 ).

(4) x^2 = A( x*B(x)*(1 + B(x)) ) where A(B(x)) = x.

a(n) ~ c * d^n / n^(3/2), where d = 3.0870367560295429... and c = 0.17761867899908... - Vaclav Kotesovec, Jul 19 2024