A247859 The product of the first n Catalan numbers and 2^(n^2).

1, 2, 32, 5120, 9175040, 197300060160, 53337309063413760, 187446932178571288903680, 8783433335287216312557974323200, 5597436690584888372318289416604667084800, 49290698636690081763273206158480893991348233830400, 6076713947745931800683801366458443411856602743866957548748800

Offset: 0

Author

Alejandro H. Morales, Sep 25 2014

Keywords

• nonn
• easy

Comments

The volume of a certain polytope (the type D_(n+2) Chan-Robbins-Yuen polytope). This was conjectured by Meszaros-Morales and proved independently by Zeilberger and Kim, both using a variant of the Morris constant term identity (just as for the original Chan-Robbins-Yuen polytope).

Links

• J. S. Kim, Proof of a conjecture of Mészáros and Morales on the volume of a flow polytope, arXiv:1407.3467, 2014.
• K. Mészáros, A. H. Morales, Flow polytopes of signed graphs and the Kostant partition function, ArXiv:1208.0140, 2012.
• D. Zeilberger, Sketch of a Proof of an Intriguing Conjecture of Karola Mészáros and Alejandro Morales Regarding the Volume of the Dn Analog of the Chan-Robbins-Yuen Polytope (Or: The Morris-Selberg Constant Term Identity Strikes Again!), arXiv:1407.2829, 2014.

Crossrefs

Cf. A000108 (Catalan numbers).

Cf. A003046 (Product of first n Catalan numbers).

Programs

Maple

seq(2^(n^2)*mul(binomial(2*k, k)/(1+k), k=0..n), n=0..13);

Mathematica

a[n_] := 2^(n^2)*Product[ CatalanNumber[k], {k, 0, n}]; Table[a[n], {n, 0, 13}]

Formula

a(n) = 2^(n^2) * A003046(n).

a(n) = 2^(n^2) * prod(k=0..n) A000108(k).