The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

%I #30 Jun 11 2021 05:05:56

%S 1,2,31,179,787,6631,2456299,33235913,158433901,17980176031,

%T 2794938616471,8546650588601,5595650767265101,35480190026972501,

%U 15523069639558351459,455264603021602214213,57023540590242398853649,949437664962426221725789,5469912218467062529961407

%N 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)!.

%D O. Furdui, Limits, Series and Fractional Part Integrals. Problems in Mathematical Analysis, Springer, New York, 2013. See Problem 3.114 and 3.118.

%H Seiichi Manyama, <a href="/A343830/b343830.txt">Table of n, a(n) for n = 1..367</a>

%H Math StackExchange, <a href="https://math.stackexchange.com/questions/2344699/compute-s-n-sum-limits-a-1-a-2-cdots-a-n-1-infty-fraca-1a-2-cdots-a-n?rq=1">Compute S_n = Sum_{a_1 a_2 ... a_n >=1} a_1 a_2 ... a_n/(a_1+a_2+...+a_n)!</a>

%F 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)!.

%F a(n) = numerator of b(n).

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

%t a[n_] := Numerator @ Sum[Binomial[n - 1, k]/(k + n)!, {k, 0, n - 1}]; Array[a, 20] (* _Amiram Eldar_, May 01 2021 *)

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

%o (PARI) a(n) = numerator(sum(k=0, n-1, binomial(n-1, k)/(k+n)!));

%Y Cf. A343831 (denominator), A343832.

%K nonn,frac

%O 1,2

%A _Seiichi Manyama_, Apr 30 2021

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 24 23:50 EDT 2024. Contains 372782 sequences. (Running on oeis4.)