A374572 - OEIS

%I #10 Jul 19 2024 05:58:45

%S 1,1,1,3,5,11,27,69,183,481,1283,3453,9361,25651,70927,197721,555039,

%T 1567345,4449023,12686465,36323203,104381397,300958959,870378337,

%U 2524129349,7338679127,21386456807,62459196233,182776933033,535861013939,1573742036447,4629306941913

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

%H Paul D. Hanna, <a href="/A374572/b374572.txt">Table of n, a(n) for n = 1..520</a>

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

%F (1) A(x)^2 = A( x*(1+x)*A(x) ).

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

%F (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 ).

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

%F a(n) ~ c * d^n / n^(3/2), where d = 3.0870367560295429... and c = 0.17761867899908... - _Vaclav Kotesovec_, Jul 19 2024

%e 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 + ...

%e where A(x)^2 = A( x*(1+x)*A(x) ).

%e RELATED SERIES.

%e 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 + ...

%e Let B(x) be the series reversion of g.f. A(x), B(A(x)) = x, then

%e 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 + ...

%e where B(x^2) = x*B(x)*(1 + B(x)).

%e SPECIFIC VALUES.

%e A(t) = 1/2 at t = 0.301949314609828865985839329094529550482897401344979...

%e where 1/4 = A( t*(1 + t)/2 ).

%e A(3/10) = 0.492388112365452715229250795508017422919418907801551...

%e where A(3/10)^2 = A( (39/100)*A(3/10) ).

%e A(2/7) = 0.443877424659041232765055763766392304444609934055603...

%e where A(2/7)^2 = A( (18/49)*A(2/7) ).

%e A(1/4) = 0.352241294433584221893793757577235288109595399125986...

%e where A(1/4)^2 = A( (5/16)*A(1/4) ).

%e A(1/5) = 0.255826785620580342641277164817159026900345909888978...

%e where A(1/5)^2 = A( (6/25)*A(1/5) ).

%o (PARI) {a(n) = my(A=[0,1], Ax); for(i=1,n, A = concat(A,0); Ax = Ser(A);

%o A[#A] = polcoeff( subst(Ax,x, x*(1+x)*Ax ) - Ax^2, #A) );A[n+1]}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A075864.

%K nonn

%O 1,4

%A _Paul D. Hanna_, Jul 18 2024