A090878 Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx.

2, 5, 26, 103, 2194, 1223, 472730, 556403, 21323986, 7281587, 125858034202, 180451625, 121437725363954, 595953719897, 26649932810926, 3211211914492699, 285050975993898158530, 549689343118061, 640611888918574971191834

Offset: 1

Author

Robert G. Wilson v, Feb 13 2004

Keywords

• nonn
• frac

Comments

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

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

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

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

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..250
• Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.
• Eric Weisstein, Exponential Sum Function

Crossrefs

Denominators are in A036505.

Cf. A120266, A063170.

Programs

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

Mathematica

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

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

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

Formula

a(n) = A036505(n-1)*Sum_{k=0..n} (A128433(n)/A128434(n)). - Reinhard Zumkeller, Mar 03 2007

Extensions

Definition corrected by Gerald McGarvey, Apr 17 2008