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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)!. 3
1, 3, 120, 2520, 51840, 2494800, 6227020800, 653837184000, 27360571392000, 30411275102208000, 51090942171709440000, 1846572624206069760000, 15511210043330985984000000, 1361108681302294020096000000, 8841761993739701954543616000000 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,frac
AUTHOR
Seiichi Manyama, Apr 30 2021
STATUS
approved

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Last modified May 29 00:29 EDT 2024. Contains 372921 sequences. (Running on oeis4.)