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 A171212 G.f.: A(x) = x + x*A(A(2x)). 3

%I #2 Mar 30 2012 18:37:20

%S 1,2,16,320,12928,985088,140861440,38451150848,20403322617856,

%T 21307854867660800,44110759073910095872,181739941085108158595072,

%U 1493546441998961207249207296,24512116566896662943648857456640

%N G.f.: A(x) = x + x*A(A(2x)).

%C More generally, if F(x) = x + x*F(F(qx)), then

%C F(x) = x + x*F(qx) + x*F(qx)*F(qF(qx) + x*F(qx)*F(qF(qx))*F(qF(qF(qx))) +...

%C with a simple solution at q=1/2:

%C F(x) = x/(1-x/2) satisfies: F(x) = x + x*F(F(x/2)).

%e G.f.: A(x) = x + 2*x^2 + 16*x^3 + 320*x^4 + 12928*x^5 +...

%e A(A(x)) = x + 4*x^2 + 40*x^3 + 808*x^4 + 30784*x^5 + 2200960*x^6 +...+ a(n)*x^n/2^(n-1) +...

%e As a formal series involving products of iterations of the g.f.,

%e A(x) = x + x*A(2x) + x*A(2x)*A(2A(2x) + x*A(2x)*A(2A(2x))*A(2A(2A(2x))) +...

%e which, upon replacing x with A(2x), yields:

%e A(A(2x)) = A(2x) + A(2x)*A(2A(2x)) + A(2x)*A(2A(2x))*A(2A(2A(2x))) +...

%e thus A(x) = x + x*A(A(2x)).

%o (PARI) {a(n,q=2)=local(A=x+x^2);for(i=1,n,A=x+x*subst(A,x,subst(A,x,q*x+O(x^n))));polcoeff(A,n)}

%Y Cf. A171213 (q=3), A171214 (q=1/3).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 08 2009

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Last modified December 10 22:20 EST 2023. Contains 367717 sequences. (Running on oeis4.)