A108552 Integer values of (1*2*...*k)/(1+2+...+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.

1, 1, 8, 180, 1120, 8064, 604800, 68428800, 830269440, 10897286400, 2324754432000, 640237370572800, 11585247657984000, 221172909834240000, 93666727314800640000, 2068161339110798131200, 47726800133326110720000, 1148978521728221184000000, 28806532937614688256000000

Offset: 1

Author

Rick L. Shepherd, Jun 09 2005

Keywords

• nonn

Comments

A000142(n)/A000217(n) = n!/(n*(n+1)/2) = 2*(n-1)!/(n+1) is an integer iff n = 1 or n + 1 is composite; i.e., iff n is a term of A060462.

Crossrefs

Cf. A060462 (corresponding k), A000142 (factorials), A000217 (triangular numbers).

Programs

Maple

select(x-> denom(x)=1, [k!/(k*(k+1)/2)$k=1..30])[];  # Alois P. Heinz, Dec 11 2020

Mathematica

Select[Table[(n - 1)!/((n (n - 1))/2), {n, 2, 50}], IntegerQ[#] &] (* Geoffrey Critzer, May 02 2015 *)

PARI

for(n=1,50, r=2*(n-1)!/(n+1); if(denominator(r)==1, print1(r,",")))

Formula

a(m) = 2*(A060462(m)-1)!/(A060462(m)+1) = A000142(A060462(m))/A000217(A060462(m)).

Extensions

Offset corrected by Alois P. Heinz, Dec 11 2020