OFFSET
1,2
REFERENCES
O. Furdui, Limits, Series and Fractional Part Integrals. Problems in Mathematical Analysis, Springer, New York, 2013. See Problem 3.114 and 3.118.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..367
Math StackExchange, Compute S_n = Sum_{a_1 a_2 ... a_n >=1} a_1 a_2 ... a_n/(a_1+a_2+...+a_n)!
FORMULA
b(n) = (1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_n / (a_1 + a_2 + ... + a_n)! = Sum_{j=0..n} (-1)^(n+j-1) * binomial(n,j) * Sum_{k=0..n+j-1} (-1)^k/k! = Sum_{k=0..n-1} binomial(n-1,k)/(k+n)!.
a(n) = numerator of b(n).
EXAMPLE
1, 2/3, 31/120, 179/2520, 787/51840, 6631/2494800, 2456299/6227020800, ...
MATHEMATICA
a[n_] := Numerator @ Sum[Binomial[n - 1, k]/(k + n)!, {k, 0, n - 1}]; Array[a, 20] (* Amiram Eldar, May 01 2021 *)
PROG
(PARI) a(n) = numerator(sum(j=0, n, (-1)^(n+j-1)*binomial(n, j)*sum(k=0, n+j-1, (-1)^k/k!)));
(PARI) a(n) = numerator(sum(k=0, n-1, binomial(n-1, k)/(k+n)!));
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Seiichi Manyama, Apr 30 2021
STATUS
approved