A206305 - OEIS

%I #33 Apr 29 2020 18:12:34

%S 1,2,56,4304,647552,161009408,59798825984,31018594543616,

%T 21421107883900928,19000925396453752832,21053097631093130264576,

%U 28496291901064818145624064,46267687031143770070546644992,88754774204916230754873248841728,198588172996269363811346709494104064

%N E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)! is inverse to f(x) = 2*x - tan(x).

%H Andrew Howroyd, <a href="/A206305/b206305.txt">Table of n, a(n) for n = 0..100</a>

%H F. Bergeron, Ph. Flajolet and B. Salvy, <a href="http://algo.inria.fr/flajolet/Publications/BeFlSa92.pdf">Varieties of Increasing Trees</a>, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.

%H D. Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions.</a> arXiv:math/0501052v2 [math.CA], 2005.

%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1211.3244">The method for obtaining expressions for coefficients of reverse generating functions</a>, arXiv:1211.3244 [math.CO], 2012.

%F a(n) = (sum(k=1..2*n, (2*n+k)!*sum(j=1..k, (-1)^(j)/(k-j)!*((sum(l=0..j-1, (1/((j-l)!*l!)*sum(m=j-l..2*n-l+j, binomial(m-1,-l+j-1)*m!*2^(2*n-m+j)*(-1)^(n+m)*Stirling2(2*n-l+j,m)))/(2*n-l+j)!)))))), n>0, a(0)=1.

%F From Peter Bala, Aug 06 2012: (Start)

%F Alternative form of the o.g.f: x/2 - (1/2)*(x - 2*atan(x))^<-1> = x/2 - (1/2)*(Integral_{t = 0..x} 1/f(t) dt)^<-1>, where f(x) = (x^2+1)/(x^2-1) and <-1> denotes series reversion w.r.t. x.

%F Applying Theorem 4.1 of Dominici to invert the integral gives the result: for n >= 1, a(n) = (1/2)*D^(2*n+1)(x) evaluated at x = 0, where D is the operator (1+x^2)/(1-x^2)*d/dx.

%F Applying Theorem 1 of Bergeron et al. gives an interpretation for the sequence as counting a variety of trees defined by the degree function phi(x) = (1+x^2)/(1-x^2) = 1 + 2*x^2 + 2*x^4 + .... We find for n >= 1 that 2*a(n) gives the number of rooted increasing plane (ordered) trees on (2*n+1) vertices with no vertices of odd outdegree and where the vertices of even outdegree come in two colors.

%F (End)

%F a(n) ~ 2^(2*n) * n^(2*n) / (exp(2*n) * (Pi/2-1)^(2*n+1/2)). - _Vaclav Kotesovec_, Jan 26 2014

%t nmax = 11; coes = CoefficientList[ x/2 - (1/2)*InverseSeries[ Series[(x - 2*ArcTan[x]), {x, 0, 2*nmax+2}], x], x]*Range[0, 2*nmax+1]!; a[n_] := coes[[2*n+2]]; Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Sep 05 2012 *)

%o (Maxima) a(n):=if n=0 then 1 else (sum((2*n+k)!*sum((-1)^(j)/(k-j)!*((sum((1/((j-l)!*l!)*sum(binomial(m-1,-l+j-1)*m!*2^(2*n-m+j)*(-1)^(n+m)*stirling2(2*n-l+j,m),m,j-l,2*n-l+j))/(2*n-l+j)!,l,0,j-1))),j,1,k),k,1,2*n));

%o (PARI) seq(n)={my(v=Vec(serlaplace(serreverse(2*x - tan(x + O(x^(2*n+2))))))); vector(n+1, i, v[2*i-1])} \\ _Andrew Howroyd_, Apr 28 2020

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Feb 06 2012

%E Terms a(12) and beyond from _Andrew Howroyd_, Apr 28 2020