%I M1987 #95 Jun 21 2024 14:31:39
%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</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 Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, <a href="https://arxiv.org/abs/2301.10830">Counting Parking Sequences and Parking Assortments Through Permutations</a>, arXiv:2301.10830 [math.CO], 2023.
%H Bernd C. Kellner, <a href="https://doi.org/10.5281/zenodo.12167556">Asymptotic products of binomial and multinomial coefficients revisited</a>, Integers 24 (2024), Article #A59, 10 pp.; arXiv:<a href="https://arxiv.org/abs/2312.11369">2312.11369</a> [math.CO], 2023.
%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="https://arxiv.org/abs/math/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 a(n) = 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 the 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
%o (PARI) a(n) = prod(k=0, n, binomial(2*k,k)/(k+1)); \\ _Michel Marcus_, Sep 06 2021
%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