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A090878 Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx. 11
2, 5, 26, 103, 2194, 1223, 472730, 556403, 21323986, 7281587, 125858034202, 180451625, 121437725363954, 595953719897, 26649932810926, 3211211914492699, 285050975993898158530, 549689343118061, 640611888918574971191834 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

FORMULA

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

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 *)

PROG

(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

(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

(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

CROSSREFS

Denominators are in A036505.

Cf. A120266, A063170.

Sequence in context: A221679 A178390 A045903 * A214951 A333004 A120762

Adjacent sequences:  A090875 A090876 A090877 * A090879 A090880 A090881

KEYWORD

nonn,frac

AUTHOR

Robert G. Wilson v, Feb 13 2004

EXTENSIONS

Definition corrected by Gerald McGarvey, Apr 17 2008

STATUS

approved

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Last modified August 3 21:16 EDT 2021. Contains 346441 sequences. (Running on oeis4.)