A218652 E.g.f. satisfies: A(x) = x + log(1 + A(x)^2).
1, 2, 12, 108, 1320, 20400, 381360, 8366400, 210712320, 5991572160, 189846961920, 6632804344320, 253310120743680, 10498203901785600, 469251125818675200, 22501933753870771200, 1152276591132072806400, 62756742945031098163200, 3622251744055050294988800
Offset: 1
Keywords
Examples
E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1320*x^5/5! +... Related series: A(x)^2 = 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1560*x^5/5! + 25200*x^6/6! +... log(1+A(x)^2) = 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1320*x^5/5! +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..375
- V. Kotesovec, Asymptotic of implicit functions if Fww = 0, Jan 19 2014
Programs
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Mathematica
Rest[CoefficientList[InverseSeries[Series[x - Log[1 + x^2],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 19 2014 *) -
PARI
{a(n)=n!*polcoeff(serreverse(x-log(1+x^2 +x*O(x^n))), n)} for(n=1, 25, print1(a(n), ", ")) -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, log(1 + x^2+x*O(x^n))^m)/m!); n!*polcoeff(A, n)} -
PARI
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, log(1 + x^2+x*O(x^n))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
Formula
E.g.f. A(x) satisfies:
(1) A(x - log(1+x^2)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) log(1+x^2)^n / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) log(1+x^2)^n/x / n! ).
a(n) = n*A218653(n-1).
a(n) ~ GAMMA(1/3) * n^(n-5/6) / (6^(1/6) * sqrt(Pi) * exp(n) * (1-log(2))^(n-1/3)). - Vaclav Kotesovec, Jan 19 2014