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

1,2

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

Cf. A343831 (denominator), A343832.

Sequence in context: A267888 A229014 A042059 * A137626 A134179 A223145

Adjacent sequences: A343827 A343828 A343829 * A343831 A343832 A343833

Keyword

nonn,frac

Author

Seiichi Manyama, Apr 30 2021

Status

approved