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A003046 Product of first n Catalan numbers.
(Formerly M1987)

%I M1987

%S 1,1,2,10,140,5880,776160,332972640,476150875200,2315045555222400,

%T 38883505145515430400,2285805733484270091494400,

%U 475475022233529990271933132800,353230394017289429773019124357120000,944693494975599542562153266945656012800000

%N Product of first n Catalan numbers.

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

%D H. W. Gould, A class of binomial sums and a series transformation, Utilitas Math., 45 (1994), 71-83.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H N. J. A. Sloane, <a href="/A003046/b003046.txt">Table of n, a(n) for n = 0..60</a>

%H V. Baldoni and M. Vergne, <a href="https://doi.org/10.1007/s00031-008-9019-8">Kostant Partitions Functions and Flow Polytopes</a>, Transform. Groups. 13 (2008), 447-469.

%H C. S. Chan et al., <a href="http://projecteuclid.org/euclid.em/1046889594">On the volume of a certain polytope">On the volume of a certain polytope</a>, Experimental Mathematics, 9 (2000), 91-99.

%H S. Corteel, J. S. Kim and K. Mészáros, <a href="https://doi.org/10.1016/j.crma.2017.01.007">Flow polytopes with Catalan volumes</a>, C. R. Math., 355 (2017), 248-259.

%H K. Mészáros and A. H. Morales, <a href="https://doi.org/10.1093/imrn/rnt212">Flow polytopes of signed graphs and the Kostant partition function</a>, IMRN 3 (2015), 830-871.

%H J. W. Moon and M. Sobel, <a href="http://dx.doi.org/10.1016/0095-8956(77)90030-2">Enumerating a class of nested group testing procedures</a>, J. Combin. Theory, B23 (1977), 184-188.

%H J. W. Moon, R. K. Guy, and N. J. A. Sloane, <a href="/A003046/a003046.pdf">Correspondence, 1973</a>

%H D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/9811108">Proof of a Conjecture of Chan, Robbins and Yuen</a>, arXiv:math/9811108 [math.CO], 1998.

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

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

%F a(n) = Sqrt[(2^n)*A069640(n)/(2*n+1)!/n! ], n>0, where A069640(n) is an inverse determinant of n X n Hilbert-like Matrix with elements M(i,j)=1/(i+j+1). - _Alexander Adamchuk_, May 17 2006

%F 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

%F 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

%p seq(mul(binomial(2*k, k)/(1+k), k=0..n), n=0..13); # _Zerinvary Lajos_, Jul 02 2008

%t a[n_] := Product[ CatalanNumber[k], {k, 0, n}]; Table[a[n], {n, 0, 13}] (* _Jean-François Alcover_, Dec 05 2012 *)

%t FoldList[Times,1,CatalanNumber[Range[20]]] (* _Harvey P. Dale_, Apr 29 2013 *)

%t 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 *)

%o (Haskell)

%o a003046 n = a003046_list !! n

%o a003046_list = scanl1 (*) a000108_list

%o -- _Reinhard Zumkeller_, Oct 01 2012

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

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_.

%E a(15) added by _Harvey P. Dale_, Apr 29 2013

%E Typo in second formula corrected by _Vaclav Kotesovec_, Nov 13 2014

%E Links added by _Alejandro H. Morales_, Jan 26 2020

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Last modified September 26 01:25 EDT 2020. Contains 337346 sequences. (Running on oeis4.)