A052716 Expansion of e.g.f. (x + 1 - sqrt(1-6*x+x^2))/2.

0, 2, 4, 36, 528, 10800, 283680, 9102240, 345058560, 15090727680, 747888422400, 41422381862400, 2535569103513600, 169983582318950400, 12386182292118835200, 974723523832041984000, 82385641026424479744000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

With a(n)=1, also number of labeled mobiles with n nodes and 2-colored internal (non-leaf) nodes - Christian G. Bower, Jun 07 2005

Links

• G. C. Greubel, Table of n, a(n) for n = 0..345
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 672
• Index entries for sequences related to mobiles

Crossrefs

Cf. A006318, A108531.

Programs

Magma

[n le 1 select 1-(-1)^n else Factorial(n)*(&+[Catalan(k)*Binomial(n+k-1, n-k-1): k in [0..n-1]]): n in [0..30]]; // G. C. Greubel, May 28 2022

Maple

spec := [S,{C=Union(B,Z),B=Prod(S,C),S=Union(Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

With[{nn=20},CoefficientList[Series[(x+1-Sqrt[1-6x+x^2])/2,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Apr 19 2020 *)

SageMath

[bool(n==1)+factorial(n)*sum(binomial(n+k-1, n-k-1)*catalan_number(k) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, May 28 2022

Formula

D-finite with recurrence: a(2)=4, a(1)=2, (n^2-1)*a(n) = (3+6*n)*a(n+1) - a(n+2).

a(n) = n!*A006318(n-1), n>=2. - R. J. Mathar, Oct 26 2013