|
%I #21 Aug 11 2014 06:39:27
%S 1,1,1,5,41,470,6804,118365,2398095,55393202,1436315357,41309995331,
%T 1305311240677,44956819853455,1676510128660807,67307814275738181,
%U 2894812673176510587,132795587656049202117,6472720746082336622865,334076240871194943910092
%N G.f. satisfies: A(x)^2 = x + A(x*A(x)^2).
%C Self-convolution yields A088223.
%H Vaclav Kotesovec, <a href="/A240996/b240996.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = B(x)^2 - x/B(x)^2 where B(x) = A(x/B(x)^2) = sqrt(x/Series_Reversion(x*A(x)^2)).
%F (2) A( x*A(x)^6 - 2*x^2*A(x)^4 + x^3*A(x)^2 ) = A(x)^4 - 3*x*A(x)^2 + x^2.
%F a(n) ~ c * 2^n * n^(n - 1/2 - log(2)/4) / (exp(n) * (log(2))^n), where c = 0.411579248322849751402... . - _Vaclav Kotesovec_, Aug 08 2014
%e G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 41*x^4 + 470*x^5 + 6804*x^6 +...
%e Compare these related series:
%e A(x)^2 = 1 + 2*x + 3*x^2 + 12*x^3 + 93*x^4 + 1032*x^5 + 14655*x^6 +...
%e A(x*A(x)^2) = 1 + x + 3*x^2 + 12*x^3 + 93*x^4 + 1032*x^5 + 14655*x^6 +...
%o (PARI) {a(n)=local(A=[1,1],Ax);for(i=1,n,A=concat(A,0);Ax=Ser(A);
%o A[#A]=Vec(1+subst(Ax,x,x*Ax^2) - Ax^2)[#A]);A[n+1]}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. A088223, A240999, A241996, A241997, A241998, A241999.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Aug 06 2014
| 