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G.f. A(x) satisfies: A( x - A(x)^2*(1 + A(x))^2 ) = x.
1

%I #9 Aug 23 2017 16:30:28

%S 1,1,6,45,414,4310,49068,598253,7707738,103981222,1459259444,

%T 21201220726,317718863636,4897066444332,77455837982360,

%U 1254882911977597,20793816009974054,351973815700006842,6079707258590589100,107070921557974264470,1921112466081500096044,35095122874748021511252,652393778217784214993656,12334667847853804120010726

%N G.f. A(x) satisfies: A( x - A(x)^2*(1 + A(x))^2 ) = x.

%C Compare g.f. to: C( x - C(x)^2*(1 - C(x))^2 ) = x, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

%H Paul D. Hanna, <a href="/A291421/b291421.txt">Table of n, a(n) for n = 1..300</a>

%e G.f.: A(x) = x + x^2 + 6*x^3 + 45*x^4 + 414*x^5 + 4310*x^6 + 49068*x^7 + 598253*x^8 + 7707738*x^9 + 103981222*x^10 + 1459259444*x^11 + 21201220726*x^12 + 317718863636*x^13 + 4897066444332*x^14 + 77455837982360*x^15 + 1254882911977597*x^16 +...

%e such that A( x - A(x)^2 - 2*A(x)^4 - A(x)^6 ) = x.

%e RELATED SERIES.

%e The g.f. of A185898 equals G(x) = A(x) + A(x)^2, which begins:

%e A(x) + A(x)^2 = x + 2*x^2 + 8*x^3 + 58*x^4 + 516*x^5 + 5264*x^6 + 59056*x^7 + 712002*x^8 + 9091360*x^9 + 121741316*x^10 +...+ A185898(n)*x^n +...

%e and satisfies G(x - G(x)^2) = x + x^2.

%e Also, we have the series:

%e A(x)^2*(1 + A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 148*x^5 + 1328*x^6 + 13520*x^7 + 150788*x^8 + 1804308*x^9 + 22852504*x^10 + 303523048*x^11 + 4199277144*x^12 +...

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

%e Define the series reversion Ai(x) by Ai(A(x)) = x, then Ai(x) begins:

%e Ai(x) = x - x^2 - 4*x^3 - 20*x^4 - 148*x^5 - 1328*x^6 - 13520*x^7 - 150788*x^8 - 1804308*x^9 - 22852504*x^10 +...

%e so that Ai(x) = x - A(x)^2*(1 + A(x))^2.

%e Finally, another series of interest is

%e sqrt(A(x) - x) = x + 3*x^2 + 18*x^3 + 153*x^4 + 1534*x^5 + 17178*x^6 + 208276*x^7 + 2685135*x^8 + 36381426*x^9 + 513935734*x^10 + 7526074612*x^11 + 113767244374*x^12 + 1769506176124*x^13 + 28247513919396*x^14 + 461885675312008*x^15 + 7723529901763157*x^16 +...

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

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

%Y Cf. A185898, A213591.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Aug 23 2017