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%I #9 Apr 17 2016 08:46:29
%S 1,2,8,38,196,1064,5988,34632,204672,1231082,7514052,46433088,
%T 289976404,1827459072,11608240000,74249294704,477826080368,
%U 3091718252320,20101537759256,131262924427560,860504352317040,5661120688863216,37363827222888640,247331149667685440,1641642515512685408,10923380539408947456,72850297774044995328,486886413558080754198,3260469757311730139044,21874082006618739609864
%N G.f. A(x) satisfies: A(x) = A( x^3 + 6*x*A(x)^3 )^(1/3), with A(0)=0, A'(0)=1.
%C Compare the g.f. to the following identity:
%C C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
%C where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%F G.f. A(x) satisfies: A( B(x) - 2*x^3/B(x) ) = x, where B(x) is described in the examples.
%e G.f.: A(x) = x + 2*x^2 + 8*x^3 + 38*x^4 + 196*x^5 + 1064*x^6 + 5988*x^7 + 34632*x^8 + 204672*x^9 + 1231082*x^10 + 7514052*x^11 + 46433088*x^12 +...
%e where A(x)^3 = A( x^3 + 6*x*A(x)^3 ).
%e RELATED SERIES.
%e A(x)^3 = x^3 + 6*x^4 + 36*x^5 + 218*x^6 + 1332*x^7 + 8208*x^8 + 50984*x^9 + 319056*x^10 + 2010528*x^11 + 12750950*x^12 + 81348948*x^13 + 521839944*x^14 + 3364421812*x^15 + 21791976192*x^16 +...
%e Series reversion of A(x) equals B(x) - 2*x^3/B(x) where
%e B(x) = x + 2*x^4 - 20*x^7 + 302*x^10 - 5436*x^13 + 108072*x^16 - 2286160*x^19 + 50475256*x^22 - 1149822240*x^25 + 26825146770*x^28 - 637700980612*x^31 + 15391872726072*x^34 - 376193675011356*x^37 + 9291840570002312*x^40 +...
%o (PARI) {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^3 + 6*X*A^3)^(1/3) ); polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A271931.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Apr 16 2016
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