A063170 -id:A063170 - cp's OEIS frontend

A134095 Expansion of e.g.f. A(x) = 1/(1 - LambertW(-x)^2).

1, 0, 2, 12, 120, 1480, 22320, 396564, 8118656, 188185680, 4871980800, 139342178140, 4363291266048, 148470651659928, 5455056815237120, 215238256785814500, 9077047768435752960, 407449611073696325536, 19396232794530856894464, 976025303642559490903980

Offset: 0

Paul D. Hanna, Oct 11 2007

nonn

E.g.f. equals the square of the e.g.f. of A060435, where A060435(n) = number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only.

			E.g.f.: A(x) = 1 + 0*x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1480*x^5/5! + ...
The formula A(x) = 1/(1 - LambertW(-x)^2) is illustrated by:
A(x) = 1/(1 - (x + x^2 + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + ...)^2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A060435; indirectly related: A062817, A132608.

Cf. A063170, A277458, A277490, A277510.

seq(simplify(GAMMA(n+1,-n)*(-exp(-1))^n),n=0..20); # Vladeta Jovovic, Oct 17 2007

CoefficientList[Series[1/(1-LambertW[-x]^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
a[x0_] := x D[1/x Exp[x], {x, n}] x^n Exp[-x] /. x->x0
Table[a[n], {n, 0, 20}] (* Gerry Martens, May 05 2016 *)

{a(n)=sum(k=0,n,(n-k)^k*k^(n-k)*binomial(n,k))}

/* Generated by e.g.f. 1/(1 - LambertW(-x)^2 ): */
{a(n)=my(LambertW=-x*sum(k=0,n,(-x)^k*(k+1)^(k-1)/k!) +x*O(x^n)); n!*polcoeff(1/(1-subst(LambertW,x,-x)^2),n)}

a(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k).
a(n) = n!*Sum_{k=0..n} (-1)^(n-k)*n^k/k!. - Vladeta Jovovic, Oct 17 2007
a(n) ~ n^n/2. - Vaclav Kotesovec, Nov 27 2012, simplified Nov 22 2021
a(n) = n! * [x^n] exp(n*x)/(1 + x). - Ilya Gutkovskiy, Sep 18 2018
a(n) = (-1)^n*exp(-n)*Integral_{x=-n..oo} x^n*exp(-x) dx. - Thomas Scheuerle, Jan 29 2024

A368951 Number of connected labeled graphs with n edges and n vertices and with loops allowed.

Original entry on oeis.org

1, 1, 2, 10, 79, 847, 11436, 185944, 3533720, 76826061, 1880107840, 51139278646, 1530376944768, 49965900317755, 1767387701671424, 67325805434672100, 2747849045156064256, 119626103584870552921, 5533218319763109888000, 270982462739224265922466

Offset: 0

Andrew Howroyd, Jan 10 2024

nonn

Exponential transform appears to be A333331. - Gus Wiseman, Feb 12 2024

			From _Gus Wiseman_, Feb 12 2024: (Start)
The a(0) = 1 through a(3) = 10 loop-graphs:
  {}  {11}  {11,12}  {11,12,13}
            {22,12}  {11,12,23}
                     {11,13,23}
                     {22,12,13}
                     {22,12,23}
                     {22,13,23}
                     {33,12,13}
                     {33,12,23}
                     {33,13,23}
                     {12,13,23}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A000169, A333331, A063170, A053506.
This is the connected covering case of A014068.
The case without loops is A057500, covering case of A370317.
Allowing any number of edges gives A062740, connected case of A322661.
This is the connected case of A368597.
The unlabeled version is A368983, connected case of A368984.
For at most n edges we have A369197.
A000085 counts set partitions into singletons or pairs.
A006129 counts covering graphs, connected A001187.
Cf. A000272, A000666, A054780, A116508, A136556, A367863, A368600.

egf:= (L-> 1-L/2-log(1+L)/2-L^2/4)(LambertW(-x)):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 10 2024

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(-log(1-t)/2 + t/2 - t^2/4 + 1))}

a(n) = A000169(n) + A057500(n) for n > 0.
E.g.f.: 1 - log(1-T(x))/2 + T(x)/2 - T(x)^2/4 where T(x) = -LambertW(-x) is the e.g.f. of A000169.
From Peter Luschny, Jan 10 2024: (Start)
a(n) = (exp(n)*Gamma(n + 1, n) - (n - 1)*n^(n - 1))/(2*n) for n > 0.
a(n) = (1/2)*(A063170(n)/n - A053506(n)) for n > 0.  (End)

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

Original entry on oeis.org

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

A120266 Numerator of Sum_{k=0..n} n^k/k!.

Original entry on oeis.org

2, 5, 13, 103, 1097, 1223, 47273, 556403, 10661993, 7281587, 62929017101, 7218065, 60718862681977, 595953719897, 13324966405463, 247016301114823, 28505097599389815853, 549689343118061, 320305944459287485595917

Offset: 1

Alexander Adamchuk, Jun 30 2006

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

			The first few fractions are 2, 5, 13, 103/3, 1097/12, 1223/5, 47273/72, 556403/315, 10661993/2240, ... = A120266/A214401. - _Petros Hadjicostas_, May 12 2020

Eric Weisstein, Exponential Sum Function.

Denominators are A214401. Cf. also A063170, A090878, A119029, A120267, A214402.

Numerator[Table[Sum[n^k/k!, {k,0,n}], {n,1,30}]]

a(n) = numerator(Sum_{k=0..n} n^k/k!).

a(n) = A063170(n)/A214402(n) = (n!/A214402(n))*Sum_{k=0..n} n^k/k! for n > 0. - Jonathan Sondow, Jul 16 2012

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

A063169 a(n) = n*A001865(n).

Original entry on oeis.org

1, 6, 51, 568, 7845, 129456, 2485567, 54442368, 1339822377, 36602156800, 1099126705611, 35986038303744, 1275815323139149, 48693140873545728, 1990581237014772375, 86778247940387209216, 4018626330009931930833, 197009947951733259436032, 10193206233792610863520867

Offset: 1

Marijke van Gans (marijke(AT)maxwellian.demon.co.uk)

Schenker sums without n-th term.
a(n)/n^n = Q(n) (called Ramanujan's function by Knuth).
Urn, n balls, with replacement: how many selections before a ball is chosen that was chosen already? Expected value is a(n)/n^n.
a(n) is the total number of recurrent elements over all endofunctions on n labeled points. a(n) = Sum_{k=1..n} A066324(n,k)*k. - Geoffrey Critzer, Dec 05 2011

			a(4) = (1*2*3*4) + 4*(2*3*4) + 4*4*(3*4) + 4*4*4*(4) = 568.

D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, Addison-Wesley, Reading, MA, 1.2.11.3 p. 116

Alois P. Heinz, Table of n, a(n) for n = 1..150
Marijke van Gans, Schenker sums
Don Zagier, Partitions, Modular Forms and Moduli Spaces, Youtube video of a lecture at Institut Henri Poincaré, Feb 28 2017.

Cf. A001865, A066324, A219706.

Flatten[Range[0, 20]! CoefficientList[Series[D[1/(1 - y t), y] /. y -> 1, {x, 0, 20}], {x, y}]]
(* Second program: *)
a[n_] := Exp[n]*Gamma[n+1, n] - n^n; Array[a, 19] (* Jean-François Alcover, Jan 25 2018 *)

a(n)=sum(k=1,n,binomial(n,k)*n^(n-k)*k!) /* Michael Somos, Jun 09 2004 */

a(n)=sum(k=1,n,binomial(n,k)*(n-k)^(n-k)*k^k) \\ Paul D. Hanna, Jul 04 2013

a(n)=sum(k=0,n-1,n!/k!*n^k) \\ Ruud H.G. van Tol, Jan 14 2023

from math import comb
def A063169(n): return (sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n) + n**n # Chai Wah Wu, Apr 25-26 2023

10 for N=1 to 42 : T=N^N : S=0
20 for K=N to 1 step -1 : T/=N : T*=K : S+=T : next K
30 print N,S : next N

a(n) = Sum_{k=0..n-1} n^k * n!/k!.
a(n)/n! = Sum_{k=0..n-1} n^k/k! (first n terms of e^n power series).
E.g.f.: T/(1-T)^2, where T=T(x) is Euler's tree function (see A000169) - Len Smiley, Nov 28 2001
E.g.f.: -LambertW(-x)/(1+LambertW(-x))^2. - Alois P. Heinz, Nov 16 2011
a(n) = A063170(n) - n^n.
a(n) = Sum_{k=1..n} C(n,k) * (n-k)^(n-k) * k^k. - Paul D. Hanna, Jul 04 2013
a(n) ~ n^(n+1/2)*sqrt(Pi/2). - Vaclav Kotesovec, Oct 05 2013
a(n) = Sum_{k=1..n} (n!/(n-k)!) * k^2 * n^(n-k-1). - Brian P Hawkins, Feb 07 2024

A119029 Numerator of Sum_{k=1..n} n^(k-1)/k!.

Original entry on oeis.org

1, 2, 4, 25, 217, 203, 6743, 69511, 1184417, 728102, 5720654791, 601499, 4670663321629, 42568060798, 888330615353, 15438515749903, 1676770323947695709, 30538296012677, 16858207434636875406943

Offset: 1

Alexander Adamchuk, Jul 22 2006

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

			The first few fractions are 1, 2, 4, 25/3, 217/12, 203/5, 6743/72, 69511/315, 1184417/2240, 728102/567, ... = A119029/A214401. - _Petros Hadjicostas_, May 12 2020

Cf. A063170, A090878, A093101, A120266, A120267, A214401 (denominators), A214402.

Numerator[Table[Sum[n^(k-1)/k!,{k,1,n}],{n,1,30}]]

a(n) = numerator(Sum_{k=1..n} n^(k-1)/k!).

a(n) = A120267(n)/n.

A120267 Numerator of Sum_{k=1..n} n^k/k!.

Original entry on oeis.org

1, 4, 12, 100, 1085, 1218, 47201, 556088, 10659753, 7281020, 62927202701, 7217988, 60718623181177, 595952851172, 13324959230295, 247016251998448, 28505095507110827053, 549689328228186, 320305941258100632731917

Offset: 1

Alexander Adamchuk, Jun 30 2006

n divides a(n) and a(n)/n = A119029(n). - Alexander Adamchuk, Oct 08 2006

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in A214401. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

			The first few fractions are 1, 4, 12, 100/3, 1085/12, 1218/5, 47201/72, 556088/315, 10659753/2240, 7281020/567, ... = A120267/A214401. - _Petros Hadjicostas_, May 12 2020

Cf. A063170, A090878, A093101, A119029, A120266, A214401 (denominators), A214402.

Numerator[Table[Sum[n^k/k!, {k,1,n}], {n,1,30}]]

a(n) = numerator(Sum_{k=1..n} n^k/k!).

a(n) = n*A119029(n). - Alexander Adamchuk, Oct 08 2006

Various sections edited by Petros Hadjicostas, May 12 2020~

A214402 Cancellation factor in reducing Sum_{k=0...n} n^k/k! to lowest terms.

Original entry on oeis.org

1, 2, 6, 8, 10, 144, 70, 128, 162, 6400, 22, 6220800, 26, 100352, 182250, 425984, 170, 429981696, 38, 163840000, 13502538, 317194240, 46, 247669456896, 31250, 1417674752, 15943230, 80564191232, 9802, 25076532510720000000, 62, 10737418240, 38196790434, 1241245548544

Offset: 1

Jonathan Sondow, Jul 15 2012

nonn

Michel Marcus, Table of n, a(n) for n = 1..300
Eric Weisstein, Exponential Sum Function.

Cf. A063170, A090878, A093101, A120266, A214401.

Table[n!/Denominator[Sum[n^k/k!, {k, 0, n}]], {n, 1, 30}]

a(n) = n!/denominator(sum(k=0, n, n^k/k!)); \\ Michel Marcus, Apr 20 2021

a(n) = n!/A214401(n).

More terms from Michel Marcus, Apr 20 2021

A214401 Denominator of Sum_{k=0..n} n^k/k!.

Original entry on oeis.org

1, 1, 1, 3, 12, 5, 72, 315, 2240, 567, 1814400, 77, 239500800, 868725, 7175168, 49116375, 2092278988800, 14889875, 3201186852864000, 14849255421, 3783802880000, 3543572316375, 562000363888803840000, 2505147019375, 496358721386591551488

Offset: 1

Jonathan Sondow, Jul 15 2012

Apparently, the three sequences T_1(n) = Sum_{k=1..n} n^(k-1)/k!, T_2(n) = Sum_{k=0..n} n^k/k!, and T_3(n) = Sum_{k=1..n} n^k/k!, with numerators in A119029, A120266, and A120267, respectively, have the same denominators, listed in the current sequence. This, however, is not immediately obvious. - Petros Hadjicostas, May 12 2020

Michel Marcus, Table of n, a(n) for n = 1..150
Eric Weisstein, Exponential Sum Function.

Numerators are A120266.

Cf. also A063170, A090878, A093101, A119029, A120267, A214402.

Denominator[Table[Sum[n^k/k!, {k, 0, n}], {n, 1, 30}]]

a(n) = denominator(sum(k=0, n, n^k/k!)); \\ Michel Marcus, Apr 20 2021

a(n) = n!/A214402(n).

cp's OEIS Frontend

A134095 Expansion of e.g.f. A(x) = 1/(1 - LambertW(-x)^2).

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A368951 Number of connected labeled graphs with n edges and n vertices and with loops allowed.

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

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A120266 Numerator of Sum_{k=0..n} n^k/k!.

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

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A063169 a(n) = n*A001865(n).

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