OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
G.f. A(x) satisfies: A( sqrt( A(2*x^2 - 16*x^3)/2 ) ) = x.
a(n) ~ c * d^n / n^(3/2), where d = 20.58647985539652206773061084116532881767... and c = 0.0190065484352102393569032... - Vaclav Kotesovec, Aug 28 2017
EXAMPLE
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 +...
such that A( 2*A(x)^2 - 16*A(x)^3 ) = 2*x^2.
RELATED SERIES.
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 +...
Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins:
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 +...
where Ai(x) = sqrt( A(2*x^2 - 16*x^3)/2 )
and Ai( 2*Ai(x)^2 ) = 2*x^2 - 16*x^3.
PROG
(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]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 21 2017
STATUS
approved