A101490 G.f. satisfies A(x) = x*(1+A^2)^2/(1-A+A^2).

0, 1, 1, 3, 8, 25, 80, 267, 911, 3170, 11192, 39993, 144320, 525124, 1924196, 7093603, 26288928, 97878831, 365918064, 1372982706, 5168555770, 19514482964, 73876936272, 280363191353, 1066357904128, 4064204607372

Offset: 0

Author

Ralf Stephan, Jan 21 2005

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..600
• M. Bousquet-Mélou, Limit laws for embedded trees
• Cyril Banderier, Markus Kuba, and Michael Wallner, Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions, arXiv:2103.03751 [math.PR], 2021.
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 413.

Programs

Maple

a:= n-> coeff(series(RootOf(A=x*(1+A^2)^2/(1-A+A^2)
        , A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 16 2013

Mathematica

For[A = 1; n = 1, n <= 26, n++, A = x*(1+A^2)^2/(1-A+A^2) + O[x]^n]; CoefficientList[A, x] (* Jean-François Alcover, Jun 29 2011, updated Apr 23 2016 *)

Maxima

T(n,k):= if n<0  then 0 else if n=k then 1 else if n>0 and k=0 then 0 else T(n-1,k)-T(n-2,k)+T(n-1,k-1)+2*T(n-3,k-1)+T(n-5,k-1);
makelist(T(2*n-1,n)/n,n,1,7); /* Vladimir Kruchinin, Sep 29 2014 */

Formula

G.f: x*c(x)*c(x^2*c(x)^2), c(x) the g.f. of A000108. - Paul Barry, Jun 02 2009

a(n+1) = Sum_{k, 0<=k<=[n/2]} A039599(n-k,k)*A000108(k). - Philippe Deléham, Apr 12 2007

a(n) ~ 2^(2*n-3/2)/(Gamma(3/4)*n^(5/4)) * (1 - Gamma(3/4)/ (n^(1/4)*sqrt(Pi/2)) + 9*Gamma(3/4)^2/(4*sqrt(2*n)*Pi)). - Vaclav Kotesovec, Sep 16 2013

a(n) = T(2*n-1,n)/n, where T(n,k)=T(n-1,k)-T(n-2,k)+T(n-1,k-1)+2*T(n-3,k-1)+T(n-5,k-1). - Vladimir Kruchinin, Sep 29 2014