A331799 Normalized volume of the Caracol flow polytope. Also equal to the number of "unified diagrams" of the Caracol graph (see Section 4.3 and Section 5 in Benedetti et al. reference).

1, 3, 32, 625, 18144, 705894, 34603008, 2051893701, 143000000000, 11464341673642, 1039964049506304, 105353940923859082, 11793014101010071552, 1445828316284179687500, 192713711798795989155840, 27750747808814680091687085, 4293818865468117678192721920

Offset: 1

Author

Alejandro H. Morales, Jan 26 2020

Keywords

• nonn
• easy

Examples

			For n=3, a(3) = 32 = 2*(3+1)^2.
		

Links

• C. Benedetti, R. S. González D'León, C. Hanusa, P. E. Harris, A. Khare, A. H. Morales, M. Yip, A combinatorial model for computing volumes of flow polytopes, arXiv:1801.07684 [math.CO], 2018-2019.
• C. Benedetti, R. S. González D'León, C. Hanusa, P. E. Harris, A. Khare, A. H. Morales, M. Yip, A combinatorial model for computing volumes of flow polytopes, Trans. Amer. Math. Soc., 372 (2019), 3369-3404.
• J. Jang and J. S. Kim, Volumes of flow polytopes related to caracol graphs, arXiv:1911.10703 [math.CO], 2019
• M. Yip, A Fuss-Catalan variation of the caracol flow polytope, arXiv:1910.10060 [math.CO], 2019.

Crossrefs

Cf. A000108, A000272, A134264.

Programs

Maple

a:=proc(n)
  return (1/n)*binomial(2*n-2,n-1)*(n+1)^(n-1);
end proc:

Mathematica

Array[(1/#) Binomial[2 # - 2, # - 1] (# + 1)^(# - 1) &, 17] (* Michael De Vlieger, Jan 28 2020 *)

Formula

a(n) = A000108(n-1)*A000272(n+1).

a(n) = (1/n)*binomial(2*n-2,n-1)*(n+1)^(n-1).

a(n) = Sum_{i>=0..n-1} binomial(2*n-2,i)*A329057(n-1,i).