A343830 a(n) = numerator of (1/e) * Sum_{a_1>=1, a_2>=1, ... , a_n>=1} a_1 * a_2 * ... * a_n / (a_1 + a_2 + ... + a_n)!.

1, 2, 31, 179, 787, 6631, 2456299, 33235913, 158433901, 17980176031, 2794938616471, 8546650588601, 5595650767265101, 35480190026972501, 15523069639558351459, 455264603021602214213, 57023540590242398853649, 949437664962426221725789, 5469912218467062529961407

Offset: 1

Author

Seiichi Manyama, Apr 30 2021

Keywords

• nonn
• frac

Examples

			1, 2/3, 31/120, 179/2520, 787/51840, 6631/2494800, 2456299/6227020800, ...
		

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)!

Crossrefs

Cf. A343831 (denominator), A343832.

Programs

Mathematica

a[n_] := Numerator @ Sum[Binomial[n - 1, k]/(k + n)!, {k, 0, n - 1}]; Array[a, 20] (* Amiram Eldar, May 01 2021 *)

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)!));

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).