A343831 a(n) = denominator 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, 3, 120, 2520, 51840, 2494800, 6227020800, 653837184000, 27360571392000, 30411275102208000, 51090942171709440000, 1846572624206069760000, 15511210043330985984000000, 1361108681302294020096000000, 8841761993739701954543616000000

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..225

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) = denominator of b(n).

Example

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

Mathematica

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

Prog

(PARI) a(n) = denominator(sum(j=0, n, (-1)^(n+j-1)*binomial(n, j)*sum(k=0, n+j-1, (-1)^k/k!)));
(PARI) a(n) = denominator(sum(k=0, n-1, binomial(n-1, k)/(k+n)!));

Crossrefs

Cf. A343830 (numerator), A343832.

Sequence in context: A316127 A317433 A214312 * A185554 A134230 A321109

Adjacent sequences: A343828 A343829 A343830 * A343832 A343833 A343834

Keyword

nonn,frac

Author

Seiichi Manyama, Apr 30 2021

Status

approved