OFFSET
1,3
COMMENTS
The denominators of the coefficients in Taylor series for LambertW(x) are 1, 1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, ..., which is this sequence prefixed by 1. (Cf. A227831.) - N. J. A. Sloane, Aug 02 2013
The second Mathematica program is faster than the first for large n. - T. D. Noe, Sep 07 2013
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., Eq. (5.66).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..200
FORMULA
a(p) = (p-1)!.
a(n) = numerator(Sum_{j = 0..n} (-1)^(n-j)*binomial(n,j)*(j/n+1)^n ). - Vladimir Kruchinin, Jun 02 2013
a(n) = denominator(n^n/n!). - Vincenzo Librandi Sep 04 2014
MAPLE
series(LambertW(x), x, 30); # N. J. A. Sloane, Jan 08 2021
MATHEMATICA
f[n_] := Select[Divisors[n! ], GCD[ #, n] == 1 &][[ -1]]; Table[f[n], {n, 30}]
Denominator[Exp[Table[Limit[Zeta[s]*Sum[(1 - If[Mod[k, n] == 0, n, 0])/k^(s - 1), {k, 1, n}], s -> 1], {n, 1, 30}]]] (* Conjecture Mats Granvik, Sep 09 2013 *)
Table[Denominator[n^n/n!], {n, 30}] (* Vincenzo Librandi, Sep 04 2014 *)
PROG
(Maxima)
a(n):=sum((-1)^(n-j)*binomial(n, j)*(j/n+1)^n, j, 0, n);
makelist(num(a(n), n, 1, 20); /* Vladimir Kruchinin, Jun 02 2013 */
(Magma) [Denominator(n^n/Factorial(n)): n in [1..25]]; // Vincenzo Librandi, Sep 04 2014
(PARI) for(n=1, 50, print1(denominator(n^n/n!), ", ")) \\ G. C. Greubel, Nov 14 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Jul 19 2004, based on a suggestion from Leroy Quet, Jun 18 2004
STATUS
approved