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%I #12 Aug 28 2017 07:15:17
%S 1,4,36,480,6896,106432,1718784,28718592,492201856,8605925376,
%T 152904727040,2752754089984,50106792767488,920624074653696,
%U 17051087289835520,318014241007730688,5967490401704681472,112584565019407941632,2134274190939740995584,40633890811539769786368,776619666947548902981632,14895370245374436645535744,286602399114033680102719488,5530627126602146509305675776,107011451193255026335799050240
%N G.f. A(x) satisfies: A(2*A(x)^2 - 16*A(x)^3) = 2*x^2.
%H Paul D. Hanna, <a href="/A291313/b291313.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies: A( sqrt( A(2*x^2 - 16*x^3)/2 ) ) = x.
%F a(n) ~ c * d^n / n^(3/2), where d = 20.58647985539652206773061084116532881767... and c = 0.0190065484352102393569032... - _Vaclav Kotesovec_, Aug 28 2017
%e G.f.: A(x) = x + 4*x^2 + 36*x^3 + 480*x^4 + 6896*x^5 + 106432*x^6 + 1718784*x^7 + 28718592*x^8 + 492201856*x^9 + 8605925376*x^10 + 152904727040*x^11 + 2752754089984*x^12 + 50106792767488*x^13 + 920624074653696*x^14 + 17051087289835520*x^15 + 318014241007730688*x^16 +...
%e such that A( 2*A(x)^2 - 16*A(x)^3 ) = 2*x^2.
%e RELATED SERIES.
%e 2*A(x)^2 - 16*A(x)^3 = 2*x^2 - 16*x^4 - 32*x^6 - 1280*x^8 - 131072*x^12 + 557056*x^14 - 22806528*x^16 + 148307968*x^18 - 5108137984*x^20 + 34520170496*x^22 +...
%e Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins:
%e Ai(x) = x - 4*x^2 - 4*x^3 - 80*x^4 - 2048*x^6 + 4352*x^7 - 89088*x^8 + 289664*x^9 - 4988416*x^10 + 16855552*x^11 - 284645376*x^12 + 1157482496*x^13 - 16504889344*x^14 + 80779878400*x^15 - 1006323073024*x^16 + 5522810216448*x^17 - 63998535434240*x^18 + 379344042950656*x^19 - 4163779031072768*x^20 +...
%e where Ai(x) = sqrt( A(2*x^2 - 16*x^3)/2 )
%e and Ai( 2*Ai(x)^2 ) = 2*x^2 - 16*x^3.
%o (PARI) {a(n) = my(V=[1]); for(i=1,n, V=concat(V,0); A = x*Ser(V); V[#V] = -polcoeff(subst(A,x, 2*A^2 - 16*A^3),#V+1)/4); V[n]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A271961, A291314, A291315.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Aug 21 2017
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