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A087047
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a(n) = n*(n+1)*(n+2)*a(n-1)/6 for n >= 1; a(0) = 1.
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3
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1, 1, 4, 40, 800, 28000, 1568000, 131712000, 15805440000, 2607897600000, 573737472000000, 164088916992000000, 59728365785088000000, 27176406432215040000000, 15218787602040422400000000, 10348775569387487232000000000, 8444600864620189581312000000000, 8182818237816963704291328000000000
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OFFSET
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0,3
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COMMENTS
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Product of the first n tetrahedral (or pyramidal) numbers. See 2nd formula. - Alexander Adamchuk, May 19 2006
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LINKS
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FORMULA
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a(n) = 2^(-n-1)*3^(-n)*n!*(n+1)!*(n+2)!.
a(n) = Product_{k=1..n} k*(k+1)*(k+2)/6.
a(n) = Product_{k=1..n} A000292(k). (End)
a(n) = denominator( [x^n] 1F3([1], [1, 2, 3], 6*x) ), where 1F3 is the hypergeometric function (see Luzón et al. at page 19). - Stefano Spezia, Oct 13 2023
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EXAMPLE
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a(4) = (1/32)*(1/81)*24*120*720 = 800.
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MAPLE
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a[0]:=1: for n from 1 to 20 do a[n]:=n*(n+1)*(n+2)*a[n-1]/6 od: seq(a[n], n=0..17); # Emeric Deutsch, Mar 06 2005
seq(mul(binomial(k+2, 3), k=1..n), n=0..16); # Zerinvary Lajos, Aug 07 2007
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MATHEMATICA
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Table[Product[k*(k+1)*(k+2)/6, {k, 1, n}], {n, 0, 16}] (* Alexander Adamchuk, May 19 2006 *)
a[n_]:=Denominator[SeriesCoefficient[HypergeometricPFQ[{1}, {1, 2, 3}, 6x], {x, 0, n}]]; Array[a, 18, 0] (* Stefano Spezia, Oct 13 2023 *)
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PROG
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(Sage)
q=50 # change q for more terms
[2^(-n-1)*3^(-n)*factorial(n)*factorial(n+1)*factorial(n+2) for n in [0..q]] # Tom Edgar, Mar 15 2014
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Enrico T. Federighi (rico125162(AT)aol.com), Aug 08 2003
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EXTENSIONS
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Example and formula corrected by Tom Edgar, Mar 15 2014
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STATUS
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approved
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