A003046 Product of first n Catalan numbers.

1, 1, 2, 10, 140, 5880, 776160, 332972640, 476150875200, 2315045555222400, 38883505145515430400, 2285805733484270091494400, 475475022233529990271933132800, 353230394017289429773019124357120000, 944693494975599542562153266945656012800000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The volume of a certain polytope (see Chan et al. reference). However, no combinatorial explanation for this is known.

References

• H. W. Gould, A class of binomial sums and a series transformation, Utilitas Math., 45 (1994), 71-83.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..60
• V. Baldoni and M. Vergne, Kostant Partitions Functions and Flow Polytopes, Transform. Groups. 13 (2008), 447-469.
• C. S. Chan et al., On the volume of a certain polytope, Experimental Mathematics, 9 (2000), 91-99.
• S. Corteel, J. S. Kim and K. Mészáros, Flow polytopes with Catalan volumes, C. R. Math., 355 (2017), 248-259.
• Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.
• Bernd C. Kellner, Asymptotic products of binomial and multinomial coefficients revisited, Integers 24 (2024), Article #A59, 10 pp.; arXiv:2312.11369 [math.CO], 2023.
• K. Mészáros and A. H. Morales, Flow polytopes of signed graphs and the Kostant partition function, IMRN 3 (2015), 830-871.
• J. W. Moon and M. Sobel, Enumerating a class of nested group testing procedures, J. Combin. Theory, B23 (1977), 184-188.
• J. W. Moon, R. K. Guy, and N. J. A. Sloane, Correspondence, 1973
• D. Zeilberger, Proof of a Conjecture of Chan, Robbins and Yuen, arXiv:math/9811108 [math.CO], 1998.
• Index to divisibility sequences

Crossrefs

Cf. A003047, A000108, A055746, A069640, A005249, A051575, A067689, A074962.

Programs

Haskell

a003046 n = a003046_list !! n
a003046_list = scanl1 (*) a000108_list
-- Reinhard Zumkeller, Oct 01 2012

Maple

seq(mul(binomial(2*k, k)/(1+k), k=0..n), n=0..13); # Zerinvary Lajos, Jul 02 2008

Mathematica

a[n_] := Product[ CatalanNumber[k], {k, 0, n}]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Dec 05 2012 *)
FoldList[Times,1,CatalanNumber[Range[20]]] (* Harvey P. Dale, Apr 29 2013 *)
Table[(2^(n^2+n-1/24) Glaisher^(3/2) BarnesG[n+3/2])/(Exp[1/8] Pi^(n/2+1/4) BarnesG[n+3]), {n, 0, 20}] (* Vladimir Reshetnikov, Nov 11 2015 *)

PARI

a(n) = prod(k=0, n, binomial(2*k,k)/(k+1)); \\ Michel Marcus, Sep 06 2021

Formula

a(n) = C(0)*C(1)*...*C(n), C() = A000108 = Catalan numbers.

a(n) = sqrt((2^n)*A069640(n)/(2*n+1)!/n!), n>0, where A069640(n) is an inverse determinant of the n X n Hilbert-like Matrix with elements M(i,j)=1/(i+j+1). - Alexander Adamchuk, May 17 2006

a(n) ~ A^(3/2) * 2^(n^2 + n - 19/24) * exp(3*n/2 - 1/8) / (n^(3*n/2 + 15/8) * Pi^(n/2+1)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014

a(n) = A^(3/2)*2^(n^2 + n - 1/24)*BarnesG(n+3/2) / (exp(1/8)*Pi^(n/2 + 1/4)*BarnesG(n+3)), where BarnesG( ) is the Barnes G-function and A is the Glaisher-Kinkelin constant (A074962). - Ilya Gutkovskiy, Mar 16 2017

For n > 0, a(n) = 2^((n+1)/2) * sqrt(BarnesG(2*n)) * Gamma(2*n) / (n^2 * (n+1) * BarnesG(n)^2 * Gamma(n)^(9/2)). - Vaclav Kotesovec, Nov 27 2024

Extensions

a(15) added by Harvey P. Dale, Apr 29 2013

Typo in second formula corrected by Vaclav Kotesovec, Nov 13 2014

Links added by Alejandro H. Morales, Jan 26 2020