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A343831
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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)!.
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3
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1, 3, 120, 2520, 51840, 2494800, 6227020800, 653837184000, 27360571392000, 30411275102208000, 51090942171709440000, 1846572624206069760000, 15511210043330985984000000, 1361108681302294020096000000, 8841761993739701954543616000000
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OFFSET
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1,2
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REFERENCES
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O. Furdui, Limits, Series and Fractional Part Integrals. Problems in Mathematical Analysis, Springer, New York, 2013. See Problem 3.114 and 3.118.
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LINKS
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FORMULA
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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).
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EXAMPLE
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1, 2/3, 31/120, 179/2520, 787/51840, 6631/2494800, 2456299/6227020800, ...
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MATHEMATICA
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a[n_] := Denominator @ Sum[Binomial[n - 1, k]/(k + n)!, {k, 0, n - 1}]; Array[a, 20] (* Amiram Eldar, May 01 2021 *)
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PROG
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(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)!));
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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