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G.f. A(x) satisfies: A( A(x)^2 - x*A(x) ) = x^3.
2

%I #15 Jun 12 2016 21:40:49

%S 1,1,-1,2,-6,16,-48,155,-506,1706,-5888,20608,-73152,262672,-951929,

%T 3478158,-12798568,47384216,-176387016,659776638,-2478574412,

%U 9347514586,-35376839998,134317287748,-511463365764,1952816800973,-7474463834606,28673987914262,-110233267218581,424608422717362,-1638541384230970,6333831090142919,-24522697340016084,95086658516947002

%N G.f. A(x) satisfies: A( A(x)^2 - x*A(x) ) = x^3.

%H Paul D. Hanna, <a href="/A272411/b272411.txt">Table of n, a(n) for n = 1..500</a>

%F If A(B(x)) = x, then g.f. A(x) and B(x) satisfy:

%F (1) A(x)^2 - x*A(x) = B(x^3).

%F (2) A(x) = x - x*C( -B(x^3)/x^2 ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

%F a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 4.06250021724219826323934729... and c = 0.03554943075321525313806189... . - _Vaclav Kotesovec_, May 03 2016

%e G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 6*x^5 + 16*x^6 - 48*x^7 + 155*x^8 - 506*x^9 + 1706*x^10 - 5888*x^11 + 20608*x^12 - 73152*x^13 + 262672*x^14 +...

%e where A( A(x)^2 - x*A(x) ) = x^3.

%e RELATED SERIES.

%e A(x)^2 = x^2 + 2*x^3 - x^4 + 2*x^5 - 7*x^6 + 16*x^7 - 48*x^8 + 158*x^9 - 506*x^10 + 1706*x^11 - 5900*x^12 + 20608*x^13 - 73152*x^14 +...

%e A(x)^2 - x*A(x) = x^3 - x^6 + 3*x^9 - 12*x^12 + 56*x^15 - 282*x^18 + 1494*x^21 - 8210*x^24 + 46365*x^27 - 267444*x^30 +...

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

%e B(x) = x - x^2 + 3*x^3 - 12*x^4 + 56*x^5 - 282*x^6 + 1494*x^7 - 8210*x^8 + 46365*x^9 - 267444*x^10 + 1568995*x^11 - 9332820*x^12 + 56156610*x^13 +...

%e such that A(x)^2 - x*A(x) = B(x^3).

%o (PARI) {a(n) = my(A=[1,1], F=x); for(i=1,n, A=concat(A,0); F=x*Ser(A); A[#A] = polcoeff(x^3 - subst(F,x, F^2 - x*F),#A+1) ); A[n]}

%o for(n=1,50,print1(a(n),", "))

%Y Cf. A273955.

%K sign

%O 1,4

%A _Paul D. Hanna_, Apr 29 2016