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A367384 Expansion of g.f. A(x) satisfying A( sqrt(A(x)^2 - 8*A(x)^3) ) = x.

%I A367384 #8 Dec 30 2023 09:34:07
%S A367384 1,2,16,172,2120,28264,396192,5746480,85394656,1291778368,19805198784,
%T A367384 306834276416,4793670528640,75415927948416,1193652980090880,
%U A367384 18994846756882176,303766882134726144,4880209392051146752,78739290124904116224,1275444751485628848128,20735204112205333970944
%N A367384 Expansion of g.f. A(x) satisfying A( sqrt(A(x)^2 - 8*A(x)^3) ) = x.
%F A367384 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A367384 (1) x = A( sqrt(A(x)^2 - 8*A(x)^3) ).
%F A367384 (2) x^2 = A(A(x))^2 - 8*A(A(x))^3, where 2*A(A(x/2)) is the g.f. of A078531.
%F A367384 (3) [x^(n+1)] A(A(x)) = 8^n * binomial((3*n-1)/2, n)/(n+1) = 2^n*A078531(n) for n >= 0.
%e A367384 G.f.: A(x) = x + 2*x^2 + 16*x^3 + 172*x^4 + 2120*x^5 + 28264*x^6 + 396192*x^7 + 5746480*x^8 + 85394656*x^9 + 1291778368*x^10 + ...
%e A367384 where A( sqrt(A(x)^2 - 8*A(x)^3) ) = x.
%e A367384 RELATED SERIES.
%e A367384 A(x)^2 = x^2 + 4*x^3 + 36*x^4 + 408*x^5 + 5184*x^6 + 70512*x^7 + 1002864*x^8 + 14711456*x^9 + 220670592*x^10 + ...
%e A367384 A(x)^3 = x^3 + 6*x^4 + 60*x^5 + 716*x^6 + 9384*x^7 + 130344*x^8 + 1882576*x^9 + 27950736*x^10 + ...
%e A367384 Let Ai(x) be the series reversion of A(x), then
%e A367384 Ai(x)^2 = A(x)^2 - 8*A(x)^3 = x^2 - 4*x^3 - 12*x^4 - 72*x^5 - 544*x^6 - 4560*x^7 - 39888*x^8 - 349152*x^9 - 2935296*x^10 - ...
%e A367384 and
%e A367384 Ai(x) = sqrt(A(x)^2 - 8*A(x)^3) = x - 2*x^2 - 8*x^3 - 52*x^4 - 408*x^5 - 3512*x^6 - 31584*x^7 - 287056*x^8 - 2560288*x^9 - ...
%e A367384 Also,
%e A367384 A(A(x)) = x + 4*x^2 + 40*x^3 + 512*x^4 + 7392*x^5 + 114688*x^6 + 1867008*x^7 + 31457280*x^8 + 543921664*x^9 + ... + 2^n*A078531(n)*x^(n+1) + ...
%e A367384 which satisfies A(A(x))^2 - 8*A(A(x))^3 = x^2, where
%e A367384 A(A(x))^2 = x^2 + 8*x^3 + 96*x^4 + 1344*x^5 + 20480*x^6 + 329472*x^7 + ...
%e A367384 A(A(x))^3 = x^3 + 12*x^4 + 168*x^5 + 2560*x^6 + 41184*x^7 + 688128*x^8 + ...
%o A367384 (PARI) {a(n) = my(A=1,V=[1]); for(i=1,n, V = concat(V,0); A = x*Ser(V);
%o A367384 V[#V] = polcoeff( x - subst(A,x, sqrt(A^2 - 8*A^3)), #V)/2 );V[n]}
%o A367384 for(n=1,30,print1(a(n),", "))
%Y A367384 Cf. A078531, A273925.
%K A367384 nonn
%O A367384 1,2
%A A367384 _Paul D. Hanna_, Dec 29 2023