This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A087047 #44 Feb 16 2025 08:32:50
%S A087047 1,1,4,40,800,28000,1568000,131712000,15805440000,2607897600000,
%T A087047 573737472000000,164088916992000000,59728365785088000000,
%U A087047 27176406432215040000000,15218787602040422400000000,10348775569387487232000000000,8444600864620189581312000000000,8182818237816963704291328000000000
%N A087047 a(n) = n*(n+1)*(n+2)*a(n-1)/6 for n >= 1; a(0) = 1.
%C A087047 Product of the first n tetrahedral (or pyramidal) numbers. See 2nd formula. - _Alexander Adamchuk_, May 19 2006
%C A087047 From _Peter Bala_, Nov 28 2024: (Start)
%C A087047 For n >= 5, a(n-3) == 9 (mod n) if and only if n is a prime (adapt the proof of the Main Theorem in Himane).
%C A087047 The list of primes p such that a(p-3) == 9 (mod p^2) (analog of A007540 - Wilson primes) begins [11, 31, 47, ...]. (End)
%H A087047 Karl Dienger, <a href="/A000217/a000217.pdf">Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung</a>, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
%H A087047 Djamel Himane, <a href="https://arxiv.org/abs/2404.08646">A simple proof of Werner Schulte's conjecture</a>, arXiv:2404.08646 [math.GM], 2024
%H A087047 Ana Luzón, Manuel A. Morón, and José L. Ramírez, <a href="https://www.researchgate.net/publication/374263654_DIFFERENTIAL_EQUATIONS_IN_WARD%27S_CALCULUS">Differential Equations in Ward's Calculus</a>, ResearchGate, September 2023.
%H A087047 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>.
%F A087047 a(n) = 2^(-n-1)*3^(-n)*n!*(n+1)!*(n+2)!.
%F A087047 From _Alexander Adamchuk_, May 19 2006: (Start)
%F A087047 a(n) = Product_{k=1..n} k*(k+1)*(k+2)/6.
%F A087047 a(n) = Product_{k=1..n} A000292(k). (End)
%F A087047 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
%e A087047 a(4) = (1/32)*(1/81)*24*120*720 = 800.
%p A087047 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
%p A087047 seq(mul(binomial(k+2, 3), k=1..n), n=0..16); # _Zerinvary Lajos_, Aug 07 2007
%t A087047 Table[Product[k*(k+1)*(k+2)/6,{k,1,n}],{n,0,16}] (* _Alexander Adamchuk_, May 19 2006 *)
%t A087047 a[n_]:=Denominator[SeriesCoefficient[HypergeometricPFQ[{1},{1,2,3},6x],{x,0,n}]]; Array[a,18,0] (* _Stefano Spezia_, Oct 13 2023 *)
%o A087047 (Sage)
%o A087047 q=50 # change q for more terms
%o A087047 [2^(-n-1)*3^(-n)*factorial(n)*factorial(n+1)*factorial(n+2) for n in [0..q]] # _Tom Edgar_, Mar 15 2014
%Y A087047 Cf. A000292, A006472.
%K A087047 nonn,easy
%O A087047 0,3
%A A087047 Enrico T. Federighi (rico125162(AT)aol.com), Aug 08 2003
%E A087047 More terms from _Emeric Deutsch_, Mar 06 2005
%E A087047 Example and formula corrected by _Tom Edgar_, Mar 15 2014
%E A087047 a(0)=1 prepended by and a(15)-a(17) from _Stefano Spezia_, Oct 13 2023