A057003 Write the natural numbers in groups: 1; 2,3; 4,5,6; 7,8,9,10; ... and multiply the members of each group.

1, 6, 120, 5040, 360360, 39070080, 5967561600, 1220096908800, 321570878428800, 106137499051584000, 42873948150095462400, 20803502274492921984000, 11938961126118491232768000, 7998487694738166709923840000, 6185983879158893706209144832000

Offset: 1

Author

Robert G. Wilson v, Sep 09 2000

Keywords

• nonn

Comments

Each group begins with a triangular number + 1 and proceeds until the next triangular number.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A006003.

Programs

Maple

a:= n-> binomial(n*(n+1)/2, n) * n!:
seq(a(n), n=1..20);  # Alois P. Heinz, Aug 17 2012

Mathematica

Table[(n (n + 1)/2)!/((n - 1) n /2)!, {n, 1, 15}]
With[{nn=15},Times@@@TakeList[Range[(nn(nn+1))/2],Range[nn]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Sep 21 2017 *)

Formula

(n (n + 1)/2)!/((n - 1) n /2)!.

a(n) ~ n^(2*n)/2^n. - Vaclav Kotesovec, Jul 13 2021