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A228867 G.f. A(x) satisfies: A(x) = -x + x^2 + A(A(x)) + Series_Reversion(A(A(x))). 0

%I

%S 1,1,8,140,3536,111118,4067460,167387778,7579673514,372383647086,

%T 19656714142024,1107002518847134,66161264225994340,

%U 4179178067931209524,278086164995822234072,19439660328872258046471,1424349675827697250143308,109166829826333936529736762

%N G.f. A(x) satisfies: A(x) = -x + x^2 + A(A(x)) + Series_Reversion(A(A(x))).

%C Compare to the Catalan function C(x) that (trivially) satisfies:

%C C(x) = -x + x^2 + C(x) + Series_Reversion(C(x)).

%e G.f.: A(x) = x + x^2 + 8*x^3 + 140*x^4 + 3536*x^5 + 111118*x^6 +...

%e where

%e A(A(x)) = x + 2*x^2 + 18*x^3 + 321*x^4 + 8144*x^5 + 256404*x^6 + 9395688*x^7 + 386935778*x^8 + 17530417564*x^9 +...

%e Let the series reversion of A(x) be denoted by G(x), which begins

%e G(x) = x - x^2 - 6*x^3 - 105*x^4 - 2658*x^5 - 83608*x^6 - 3062080*x^7 -...

%e then

%e G(G(x)) = x - 2*x^2 - 10*x^3 - 181*x^4 - 4608*x^5 - 145286*x^6 - 5328228*x^7 - 219548000*x^8 - 9950744050*x^9 +...

%e so that A(x) = -x + x^2 + A(A(x)) + G(G(x)).

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

%o for(n=1,30,print1(a(n),", "))

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 06 2013

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)