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E.g.f: A(x) = f(x*A(x)^3), where f(x) = (1+4*x)*exp(x).
1, 5, 159, 10228, 1009253, 135069696, 22882888555, 4696799559488, 1133128780421385, 314294095403352064, 98550149514670698071, 34473870245560804316160, 13310522831484403851847981, 5622806397207798234900070400, 2579680348909056700728913816227
a(n) equals the coefficient of = n! * [x^n in ] ((1+4*x)*exp(x))^(3*n+1)/(3*n+1).
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a(n) ~ 16^(2*n+1) * n^(n-1) / (sqrt(13) * 9^(n+1) * exp(3*n/4 - 1/12)). - Vaclav Kotesovec, Jan 24 2014
Table[n!*SeriesCoefficient[((1+4*x)*E^x)^(3*n+1)/(3*n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 24 2014 *)
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_Paul D. Hanna (pauldhanna(AT)juno.com), _, Oct 07 2003
Radius of convergence of A(x): r = (3^2/4^4)*exp(-1/4) = 0.0273797..., where A(r) = (4/3)*exp(1/12), and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.
nonn,new
nonn
nonn,new
nonn
Paul D . Hanna (pauldhanna(AT)juno.com), Oct 07 2003
E.g.f: A(x) = f(x*A(x)^3), where f(x)=(1+4*x)*exp(x).
1, 5, 159, 10228, 1009253, 135069696, 22882888555, 4696799559488, 1133128780421385, 314294095403352064, 98550149514670698071, 34473870245560804316160, 13310522831484403851847981
0,2
Radius of convergence of A(x): r = (3^2/4^4)*exp(-1/4) = 0.0273797..., where A(r) = (4/3)*exp(1/12), and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.
a(n) equals the coefficient of x^n in ((1+4*x)*exp(x))^(3*n+1)/(3*n+1).
(PARI) a(n)=n!*polcoeff(((1+4*x)*exp(x))^(3*n+1)+x*O(x^n), n, x)/(3*n+1)
nonn
Paul D Hanna (pauldhanna(AT)juno.com), Oct 07 2003
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