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 A003046 Product of first n Catalan numbers. (Formerly M1987) 19
 1, 1, 2, 10, 140, 5880, 776160, 332972640, 476150875200, 2315045555222400, 38883505145515430400, 2285805733484270091494400, 475475022233529990271933132800, 353230394017289429773019124357120000, 944693494975599542562153266945656012800000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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 C. S. Chan et al., On the volume of a certain polytope">On the volume of a certain polytope, Experimental Mathematics, 9 (2000), 91-99. J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002; see Conjecture 4. 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. FORMULA 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 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 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 *) PROG (Haskell) a003046 n = a003046_list !! n a003046_list = scanl1 (*) a000108_list -- Reinhard Zumkeller, Oct 01 2012 CROSSREFS Cf. A003047, A000108, A055746, A069640, A005249, A051575, A067689, A074962. Sequence in context: A014228 A059475 A156296 * A294115 A137884 A057565 Adjacent sequences:  A003043 A003044 A003045 * A003047 A003048 A003049 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS a(15) added by Harvey P. Dale, Apr 29 2013 Typo in second formula corrected by Vaclav Kotesovec, Nov 13 2014 STATUS approved

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Last modified December 14 05:17 EST 2018. Contains 318090 sequences. (Running on oeis4.)