A087047 a(n) = n*(n+1)*(n+2)*a(n-1)/6 for n >= 1; a(0) = 1.

1, 1, 4, 40, 800, 28000, 1568000, 131712000, 15805440000, 2607897600000, 573737472000000, 164088916992000000, 59728365785088000000, 27176406432215040000000, 15218787602040422400000000, 10348775569387487232000000000, 8444600864620189581312000000000, 8182818237816963704291328000000000

Offset

0,3

Comments

Product of the first n tetrahedral (or pyramidal) numbers. See 2nd formula. - Alexander Adamchuk, May 19 2006

From Peter Bala, Nov 28 2024: (Start)

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).

The list of primes p such that a(p-3) == 9 (mod p^2) (analog of A007540 - Wilson primes) begins [11, 31, 47, ...]. (End)

Links

Table of n, a(n) for n=0..17.

Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Djamel Himane, A simple proof of Werner Schulte's conjecture, arXiv:2404.08646 [math.GM], 2024

Ana Luzón, Manuel A. Morón, and José L. Ramírez, Differential Equations in Ward's Calculus, ResearchGate, September 2023.

Eric Weisstein's World of Mathematics, Tetrahedral Number.

Formula

a(n) = 2^(-n-1)*3^(-n)*n!*(n+1)!*(n+2)!.

From Alexander Adamchuk, May 19 2006: (Start)

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

Example

a(4) = (1/32)*(1/81)*24*120*720 = 800.

Maple

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000292, A006472.

Sequence in context: A277748 A278590 A370086 * A211035 A053514 A325293

Adjacent sequences: A087044 A087045 A087046 * A087048 A087049 A087050

Keyword

nonn,easy

Author

Enrico T. Federighi (rico125162(AT)aol.com), Aug 08 2003

Extensions

More terms from Emeric Deutsch, Mar 06 2005

Example and formula corrected by Tom Edgar, Mar 15 2014

a(0)=1 prepended by and a(15)-a(17) from Stefano Spezia, Oct 13 2023

Status

approved