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 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)!. 3
 1, 2, 31, 179, 787, 6631, 2456299, 33235913, 158433901, 17980176031, 2794938616471, 8546650588601, 5595650767265101, 35480190026972501, 15523069639558351459, 455264603021602214213, 57023540590242398853649, 949437664962426221725789, 5469912218467062529961407 (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 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

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Last modified April 13 12:21 EDT 2024. Contains 371641 sequences. (Running on oeis4.)