A183611 E.g.f. satisfies: A'(x) = A(x)^2 + x*A(x)^3, with A(0) = 1.
1, 1, 3, 14, 91, 756, 7657, 91504, 1260441, 19663280, 342669691, 6597811584, 139094618467, 3186675803584, 78834061767825, 2094418664339456, 59474007876381553, 1797637447068293376, 57623116235327599411
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 14*x^3/3! + 91*x^4/4! +... A'(x) = 1 + 3*x + 14*x^2/2! + 91*x^3/3! + 756*x^4/4! +... A(x)^2 = 1 + 2*x + 8*x^2/2! + 46*x^3/3! + 348*x^4/4! + 3262*x^5/5! +... A(x)^3 = 1 + 3*x + 15*x^2/2! + 102*x^3/3! + 879*x^4/4! + 4395*x^5/5! +... E.g.f. A(x) = d/dx Series_Reversion(G(x)) where G(x) begins: G(x) = x - x^2/2! + x^4/4! - x^5/5! + x^7/7! - x^8/8! + x^10/10! - x^11/11! +... The series reversion of G(x) begins: x + x^2/2! + 3*x^3/3! + 14*x^4/4! + 91*x^5/5! + 756*x^6/6! +...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..400
- V. Dotsenko, Pattern avoidance in labelled trees, arXiv preprint arXiv:1110.0844 [math.CO], 2011-2012.
Programs
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Mathematica
terms = 20; A[_] = 0; Do[A[x_] = 1+Integrate[A[x]^2 + x A[x]^3, x]+O[x]^terms // Normal, terms]; CoefficientList[A[x], x] Range[0, terms-1]! (* Jean-François Alcover, Oct 27 2018 *)
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PARI
{a(n)=local(A=1);for(n=0,n,A=1+A*intformal(1+x*A+x*O(x^n)));n!*polcoeff(A,n)} -
PARI
{a(n)=n!*polcoeff(deriv(serreverse(sum(m=1,n\3+1,x^(3*m-2)/(3*m-2)!-x^(3*m-1)/(3*m-1)!+x^2*O(x^n)))),n)}
Formula
E.g.f.: A(x) = 1 + A(x)*[Integral 1 + x*A(x) dx], where the integration does not include the constant term.
E.g.f.: d/dx Series_Reversion(Sum_{n>=1} x^(3*n-2)/(3*n-2)! - x^(3*n-1)/(3*n-1)!).
a(n) ~ n^n * exp(Pi*(n+1)/(3*sqrt(3))-n). - Vaclav Kotesovec, Feb 19 2014