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A003046
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Product of first n Catalan numbers.
(Formerly M1987)
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14
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1, 1, 2, 10, 140, 5880, 776160, 332972640, 476150875200, 2315045555222400, 38883505145515430400, 2285805733484270091494400, 475475022233529990271933132800, 353230394017289429773019124357120000, 944693494975599542562153266945656012800000
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OFFSET
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0,3
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COMMENTS
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The volume of a certain polytope (see Chan et al. reference). However, no combinatorial explanation for this is known.
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REFERENCES
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C. S. Chan et al., On the volume of a certain polytope, Experimental Mathematics, 9 (2000), 91-99.
H. W. Gould, A class of binomial sums and a series transformation, Utilitas Math., 45 (1994), 71-83.
J. W. Moon and M. Sobel, Enumerating a class of nested group testing procedures, J. Combin. Theory, B23 (1977), 184-188.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..60
D. Zeilberger, [math/9811108] Proof of a Conjecture of Chan, Robbins and Yuen
Index to divisibility sequences
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FORMULA
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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
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MAPLE
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seq(mul(binomial(2*k, k)/(1+k), k=0..n), n=0..13); - Zerinvary Lajos, Jul 02 2008
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a003046 n = a003046_list !! n
a003046_list = scanl1 (*) a000108_list
-- Reinhard Zumkeller, Oct 01 2012
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CROSSREFS
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Cf. A003047, A000108, A055746, A069640, A005249, A067689, A074962.
Sequence in context: A014228 A059475 A156296 * A137884 A057565 A152804
Adjacent sequences: A003043 A003044 A003045 * A003047 A003048 A003049
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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a(15) added by Harvey P. Dale, Apr 29 2013
Typo in second formula corrected by Vaclav Kotesovec, Nov 13 2014
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STATUS
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approved
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