A036503 -id:A036503 - cp's OEIS frontend

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

Robert G. Wilson v, Feb 13 2004

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

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

Denominators are in A036505.

Cf. A120266, A063170.

[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

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 *)

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

[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

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

Definition corrected by Gerald McGarvey, Apr 17 2008

A095996 a(n) = largest divisor of n! that is coprime to n.

Original entry on oeis.org

1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, 479001600, 868725, 14350336, 638512875, 20922789888000, 14889875, 6402373705728000, 14849255421, 7567605760000, 17717861581875, 1124000727777607680000, 2505147019375

Offset: 1

Robert G. Wilson v, Jul 19 2004, based on a suggestion from Leroy Quet, Jun 18 2004

nonn

The denominators of the coefficients in Taylor series for LambertW(x) are 1, 1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, ..., which is this sequence prefixed by 1. (Cf. A227831.) - N. J. A. Sloane, Aug 02 2013

The second Mathematica program is faster than the first for large n. - T. D. Noe, Sep 07 2013

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., Eq. (5.66).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A036503, A227831, A066570.

[Denominator(n^n/Factorial(n)): n in [1..25]]; // Vincenzo Librandi, Sep 04 2014

series(LambertW(x),x,30); # N. J. A. Sloane, Jan 08 2021

f[n_] := Select[Divisors[n! ], GCD[ #, n] == 1 &][[ -1]]; Table[f[n], {n, 30}]
Denominator[Exp[Table[Limit[Zeta[s]*Sum[(1 - If[Mod[k, n] == 0, n, 0])/k^(s - 1), {k, 1, n}], s -> 1], {n, 1, 30}]]] (* Conjecture Mats Granvik, Sep 09 2013 *)
Table[Denominator[n^n/n!], {n, 30}] (* Vincenzo Librandi, Sep 04 2014 *)

a(n):=sum((-1)^(n-j)*binomial(n,j)*(j/n+1)^n,j,0,n);
makelist(num(a(n)),n,1,20); /* Vladimir Kruchinin, Jun 02 2013 */

a(n) = denominator(n^n/n!); \\ G. C. Greubel, Nov 14 2017

a(p) = (p-1)!.
a(n) = n!/A051696(n) = (n-1)!/A062763(n).
a(n) = numerator(Sum_{j = 0..n} (-1)^(n-j)*binomial(n,j)*(j/n+1)^n ). - Vladimir Kruchinin, Jun 02 2013
a(n) = denominator(n^n/n!). - Vincenzo Librandi Sep 04 2014

A036505 Numerator of (n+1)^n/n!.

Original entry on oeis.org

1, 2, 9, 32, 625, 324, 117649, 131072, 4782969, 1562500, 25937424601, 35831808, 23298085122481, 110730297608, 4805419921875, 562949953421312, 48661191875666868481, 91507169819844, 104127350297911241532841, 640000000000000000, 865405750887126927009

Offset: 0

N. J. A. Sloane

Also denominator of Sum_{k=0..n} binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) [Prodinger]. - N. J. A. Sloane, Jul 31 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.

Cf. A036503, A063170.

Cf. A095996 (denominators).

List([0..20], n -> NumeratorRat((n+1)^n/Factorial(n))); # Muniru A Asiru, Feb 12 2018

[Numerator((n+1)^n/Factorial(n)): n in [0..20]]; // Vincenzo Librandi, Sep 10 2013

a:=n -> numer((n+1)^n/factorial(n)):  A036505 := [seq(a(n), n=0..20)]; # Muniru A Asiru, Feb 12 2018

CoefficientList[Series[1/(1 + ProductLog[-x]), {x, 0, 21}], x] // Numerator // Rest (* Jean-François Alcover, Feb 04 2013, after Vladimir Kruchinin *)

my(x='x+O('x^30)); apply(x -> numerator(x), Vec(-1+1/(1+lambertw(-x)))) \\ G. C. Greubel and Michel Marcus, Feb 08 2019

[numerator((n+1)^n/factorial(n)) for n in (0..20)] # G. C. Greubel, Feb 08 2019

a(n) = A090878(n+1)/Sum_{k=0..n+1} (A128433(n+1)/A128434(n+1)). - Reinhard Zumkeller, Mar 03 2007
G.f.: -x*e^(-LambertW(-x))/((LambertW(-x)+1)*LambertW(-x)). - Vladimir Kruchinin, Feb 04 2013
A simpler g.f. is 1/(1 + LambertW(-x)). - Jean-François Alcover, Feb 04 2013

A036504 Numerator of n^(n-1)/n!.

Original entry on oeis.org

1, 1, 3, 8, 125, 54, 16807, 16384, 531441, 156250, 2357947691, 2985984, 1792160394037, 7909306972, 320361328125, 35184372088832, 2862423051509815793, 5083731656658, 5480386857784802185939, 32000000000000000, 41209797661291758429

Offset: 1

N. J. A. Sloane

(-1)^(n+1)* n-th numerator of coefficient in LambertW(x) power series, where LambertW(x) is the transcendental function satisfying LambertW(x)*exp( LambertW(x) )=x. - Benoit Cloitre, May 08 2002

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A036503 (denominators).

A095996 is a better version of the denominators of the LambertW function. - N. J. A. Sloane, Jan 08 2021

[Numerator(n^(n-1)/Factorial(n)): n in [1..30]]; // G. C. Greubel, Sep 09 2018

Numerator[Table[n^(n - 1)/n!, {n, 1, 30}]] (* G. C. Greubel, Sep 09 2018 *)
CoefficientList[-LambertW[-x]/x + O[x]^21, x] // Numerator (* Jean-François Alcover, Jan 15 2019 *)

for(n=1, 30, print1(numerator(n^(n-1)/n!), ", ")) \\ G. C. Greubel, Sep 09 2018

[numerator(n^(n-1)/factorial(n)) for n in (1..30)] # G. C. Greubel, Feb 08 2019

A227831 Numerators of coefficients in Taylor series for LambertW(x).

Original entry on oeis.org

0, 1, -1, 3, -8, 125, -54, 16807, -16384, 531441, -156250, 2357947691, -2985984, 1792160394037, -7909306972, 320361328125, -35184372088832, 2862423051509815793, -5083731656658, 5480386857784802185939, -32000000000000000, 41209797661291758429, -244636361793658185164

Offset: 0

N. J. A. Sloane, Aug 01 2013

The denominators are 1, 1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, ..., which is A095996 prefixed by 1.

			0, 1, -1, 3/2, -8/3, 125/24, -54/5, 16807/720, -16384/315, 531441/4480, -156250/567, 2357947691/3628800, -2985984/1925, ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., Eq. (5.66).
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 34.

Alois P. Heinz, Table of n, a(n) for n = 0..300
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On the Lambert W Function
Eric Weisstein's World of Mathematics, Lambert W-Function
Wikipedia, Lambert W function

Cf. A095996. See also A036504/A036503.

series(LambertW(x),x,30); # N. J. A. Sloane, Jan 08 2021

Numerator[CoefficientList[Series[LambertW[x], {x, 0, 22}], x]] (* Mats Granvik, Nov 24 2013 *)
Numerator[CoefficientList[InverseSeries[Series[x/Sum[((-x)^n)/Factorial[n], {n, 0, 22}], {x, 0, 22}]], x]] (* Mats Granvik, Nov 24 2013 *)

Numerators of series reversion of x/(Sum_{n=0..infinity} ((-x)^n)/n!). - Mats Granvik, Nov 24 2013

A036502 Numerator of n^(n-2)/n!.

Original entry on oeis.org

1, 1, 1, 2, 25, 9, 2401, 2048, 59049, 15625, 214358881, 248832, 137858491849, 564950498, 21357421875, 2199023255552, 168377826559400929, 282429536481, 288441413567621167681, 1600000000000000, 1962371317204369449, 11119834626984462962

Offset: 1

N. J. A. Sloane

			1, 1/2, 1/2, 2/3, 25/24, 9/5, 2401/720, 2048/315, 59049/4480, 15625/567, 214358881/3628800, ...

Cf. A036503, A036504.

cp's OEIS Frontend

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

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A095996 a(n) = largest divisor of n! that is coprime to n.

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A036505 Numerator of (n+1)^n/n!.

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A036504 Numerator of n^(n-1)/n!.

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A227831 Numerators of coefficients in Taylor series for LambertW(x).

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Maple

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A036502 Numerator of n^(n-2)/n!.

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