A078483 G.f.: -2*x/(1 - 5*x - sqrt(1-4*x) + x*sqrt(1-4*x) + 2*x^2).
1, 1, 2, 6, 20, 69, 243, 869, 3145, 11491, 42312, 156807, 584288, 2187298, 8221257, 31009841, 117331070, 445174418, 1693270531, 6454992143, 24657428519, 94363587324, 361741068087, 1388892123038, 5340282880156, 20560742443041, 79259430563491, 305889059254747
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Preprint, 2002.
- M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Discrete Math., 259 (2002), 19-36.
Programs
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Mathematica
catGF = (1 - Sqrt[1 - 4 x])/(2 x); CoefficientList[Normal[Series[1/(1 - (x + x^2 catGF^3)), {x, 0, 20}]], x] (* David Callan, Feb 06 2016 *) CoefficientList[Series[-2 x / (1 - 5 x - Sqrt[1 - 4 x] + x Sqrt[1 - 4 x] + 2 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, May 28 2016 *) -
Maxima
a(n):=sum(m*sum(((sum(binomial(j,-2*m-k+2*j)*binomial(m+k,j),j,0,m+k))*binomial(n-m-1,k-1))/(m+k),k,1,n-m),m,1,n)+1; /* Vladimir Kruchinin, Oct 11 2011 */
Formula
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) is the upper left term in M^n, where M is the following infinite square production matrix:
1, 1, 0, 0, 0, 0, ...
1, 2, 1, 0, 0, 0, ...
1, 1, 1, 1, 0, 0, ...
1, 1, 1, 1, 1, 0, ...
1, 1, 1, 1, 1, 1, ...
... (End)
a(n) = 1 + Sum_{m=1..n} m*Sum_{k=1..n-m} (1/(m+k)) * ((Sum_{j=0..m+k} binomial(j,-2*m-k+2*j)*binomial(m+k,j))*binomial(n-m-1,k-1)). - Vladimir Kruchinin, Oct 11 2011
G.f.: 1/(1 - (x + x^2 * C(x)^3)) where C(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. for the Catalan numbers A000108. - David Callan, Feb 06 2016
a(n) ~ 3 * 2^(2*n + 2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 20 2019
D-finite with recurrence: n*a(n) +2*(-5*n+4)*a(n-1) +3*(11*n-18)*a(n-2) +(-41*n+102)*a(n-3) +(21*n-64)*a(n-4) +2*(-2*n+7)*a(n-5)=0. - R. J. Mathar, Jan 23 2020
Extensions
Replaced definition with g.f. given by Atkinson and Still (2002). - N. J. A. Sloane, May 24 2016
Comments