A090878 - OEIS

%I #27 Sep 08 2022 08:45:12

%S 2,5,26,103,2194,1223,472730,556403,21323986,7281587,125858034202,

%T 180451625,121437725363954,595953719897,26649932810926,

%U 3211211914492699,285050975993898158530,549689343118061,640611888918574971191834

%N Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx.

%C Also numerators of e_n(n) where e_n(x) is the exponential sum function exp_n(x) and where denominators are given by either A095996 (largest divisor of n! that is coprime to n) or A036503 (denominator of n^(n-2)/n!). - _Gerald McGarvey_, Nov 14 2005

%C a(n) is a multiple of A120266(n) or equals A120266(n), A120266(n) is numerator of Sum_{k=0..n} n^k/k!, the integral = (n-1)!/n^(n-1) * the Sum. - _Gerald McGarvey_, Apr 17 2008

%C The integral = (1/n^n)*A063170[n] (Schenker sums with n-th term, Integral_{x>0} exp(-x)*(n+x)^n dx). - _Gerald McGarvey_, Apr 17 2008

%C Expected value in the birthday paradox problem.  Let X be a random variable that assigns to each f:{1,2,...,n+1}->{1,2,...,n} the smallest k in {2,3,...,n+1} such that f(k)=f(j) for some j < k.  a(n)/A036505(offset=1) = E(X) the expected value of X.  For n=365 E(X) is (surprising low) approximately 24. - _Geoffrey Critzer_, May 18 2013

%C Also numerator of Sum_{k=0..n} binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) [Prodinger]. _N. J. A. Sloane_, Jul 31 2013

%H G. C. Greubel, <a href="/A090878/b090878.txt">Table of n, a(n) for n = 1..250</a>

%H Helmut Prodinger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i3p7/0">An identity conjectured by Lacasse via the tree function</a>, Electronic Journal of Combinatorics, 20(3) (2013), #P7.

%H Eric Weisstein, <a href="http://mathworld.wolfram.com/ExponentialSumFunction.html">Exponential Sum Function</a>

%F a(n) = A036505(n-1)*Sum_{k=0..n} (A128433(n)/A128434(n)). - _Reinhard Zumkeller_, Mar 03 2007

%t f[n_]:= Integrate[E^(-x)*(1+x/n)^n, {x,0,Infinity}]; Table[Numerator[ f[n]], {n, 1, 20}]

%t Table[Numerator[1 + Sum[If[k==0,1,Binomial[n,k]*(k/n)^k*((n-k)/n)^(n-k)], {k,0,n-1}]], {n,1,20}] (* _G. C. Greubel_, Feb 08 2019 *)

%o (PARI) vector(20, n, numerator(sum(k=0, n, binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k)))) \\ _G. C. Greubel_, Feb 08 2019

%o (Magma) [Numerator((&+[Binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k): k in [0..n]])): n in [1..20]]; // _G. C. Greubel_, Feb 08 2019

%o (Sage) [numerator(sum(binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) for k in (0..n))) for n in (1..20)] # _G. C. Greubel_, Feb 08 2019

%Y Denominators are in A036505.

%Y Cf. A120266, A063170.

%K nonn,frac

%O 1,1

%A _Robert G. Wilson v_, Feb 13 2004

%E Definition corrected by _Gerald McGarvey_, Apr 17 2008