A277458 -id:A277458 - 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

A133297 a(n) = n!Sum_{k=1..n} (-1)^(k+1)n^(n-k-1)/(n-k)!.

Original entry on oeis.org

0, 1, 1, 5, 34, 329, 4056, 60997, 1082320, 22137201, 512801920, 13269953861, 379400765184, 11877265764025, 404067857880064, 14843708906336325, 585606019079612416, 24693567694861202273, 1108343071153648926720, 52757597474618636748421, 2654611611461360017408000

Offset: 0

Vladeta Jovovic, Oct 17 2007

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A001865 (Gamma(n, n)/exp(-n)).

Cf. A000272, A000435, A277458, A292977.

a:= function(n)
    if n=0 then return 0;
    else return Factorial(n)*Sum([1..n], k-> (-1)^(k+1)*n^(n-k-1)/Factorial(n-k));
    fi;
  end;
List([0..25], n-> a(n) ); # G. C. Greubel, Aug 02 2019

a:= func< n | n eq 0 select 0 else Factorial(n)*(&+[(-1)^(k+1)*n^(n-k-1)/Factorial(n-k): k in [1..n]]) >;
[a(n): n in [0..25]]; // G. C. Greubel, Aug 02 2019

Table[n!*Sum[(-1)^(k+1)*n^(n-k-1)/(n-k)!, {k,n}], {n,0,25}] (* Stefan Steinerberger, Oct 19 2007 *)
With[{m=25}, CoefficientList[Series[Log[1-LambertW[-x]], {x,0,m}], x]*Range[0,m]!] (* G. C. Greubel, Aug 02 2019 *)

my(x='x+O('x^25)); concat([0], Vec(serlaplace( log(1-lambertw(-x)) ))) \\ G. C. Greubel, Aug 02 2019

def a(n):
    if (n==0): return 0
    else: return factorial(n)*sum((-1)^(k+1)*n^(n-k-1)/factorial(n-k) for k in (1..n))
[a(n) for n in (0..25)] # G. C. Greubel, Aug 02 2019

E.g.f.: log(1-LambertW(-x)).
a(n) ~ n^(n-1)/2. - Vaclav Kotesovec, Sep 25 2013
Conjecture: a(n) = (n-1)!*( Sum_{k >= 0} (-1)^k * n^(n+k)/(n+k)! - (-1/e)^n ) for n >= 1. Cf. A000435. - Peter Bala, Jul 23 2021
From Thomas Scheuerle, Nov 17 2023: (Start)
This conjecture is true. Let "gamma" be the lower incomplete gamma function: gamma(n, x) = (n-1)! (1 - exp(-x)*Sum_{k = 0..n-1} x^k/k! ), then we can get the upper incomplete gamma function Gamma(n, x) = gamma(n, oo) - gamma(n, x). By inserting according the formula below, we will obtain the formula from Peter Bala.
a(n) = (-1)^(n+1)*Gamma(n, -n)/exp(n) = (-1)^(n+1)*A292977(n-1, n), for n > 0, where Gamma is the upper incomplete gamma function. (End)

More terms from Stefan Steinerberger, Oct 19 2007

A277510 E.g.f.: -1/(1-LambertW(-x))^2.

Original entry on oeis.org

-1, 2, -2, 6, 8, 170, 1872, 29246, 519808, 10698642, 248787200, 6458737142, 185138721792, 5808233422394, 197952647108608, 7283047491096750, 287705410381709312, 12145740050403520034, 545696709922799419392, 25998534614835587104742, 1309210567403228200960000

Offset: 0

Vaclav Kotesovec, Oct 18 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A063170, A134095, A277458, A277490, A308506.

CoefficientList[Series[-1/(1-LambertW[-x])^2, {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^50); Vec(serlaplace(-1/(1 - lambertw(-x))^2)) \\ G. C. Greubel, Nov 08 2017

a(n) ~ n^(n-1) / 4.

A308506 Expansion of e.g.f.: -1/(1-LambertW(-2*x)).

Original entry on oeis.org

-1, 2, 0, 24, 256, 5280, 129024, 3893120, 138215424, 5657154048, 262183321600, 13572739749888, 776265384591360, 48609716407476224, 3307818108252585984, 243052603284860928000, 19179014510218162733056, 1617564760662882792898560, 145212699111541646687207424

Offset: 0

Vaclav Kotesovec, Jun 02 2019

sign

Robert Israel, Table of n, a(n) for n = 0..351

Cf. A000312, A062971, A277458.

de:= diff(y(x),x) = x*y(x)^3/(1-2*x*y(x)):
S:= rhs(dsolve({de, y(0)=2},y(x), series, order=40)):
-1, seq(coeff(S,x,i)*(i+1)!,i=0..39); # Robert Israel, Apr 13 2020

CoefficientList[Series[-1/(1-LambertW[-2*x]), {x, 0, 20}], x] * Range[0, 20]!

my(x='x+O('x^20)); Vec(serlaplace(-1/(1-lambertw(-2*x)))) \\ Michel Marcus, Apr 13 2020

a(n) ~ 2^(n-2) * n^(n-1).

A332048 a(n) = n! * [x^n] 1 / (1 - LambertW(x))^n.

Original entry on oeis.org

1, 1, 2, 15, 104, 1145, 13824, 208831, 3536000, 68918769, 1489702400, 35742514511, 937323767808, 26750313223465, 824073079660544, 27276657371589375, 965004380380626944, 36347144974616190689, 1451974448007830568960, 61319892272079181137679, 2729671240750270054400000

Offset: 0

Ilya Gutkovskiy, Feb 06 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..394

Cf. A239761, A277458, A277510.

Table[n! SeriesCoefficient[1/(1 - LambertW[x])^n, {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[Sum[Sum[(-1)^(n - k) Binomial[n - 1, j] StirlingS1[j + 1, k] n^(n + k - j - 1), {j, 0, n - 1}], {k, 0, n}], {n, 1, 20}]]

a(n) = Sum_{k=0..n} Sum_{j=0..n-1} (-1)^(n - k) * binomial(n - 1, j) * Stirling1(j + 1, k) * n^(n + k - j - 1) for n > 0.

a(n) ~ phi^(3*n + 1/2) * n^n / (5^(1/4) * exp(n + n/phi)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Feb 07 2020

A347993 a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * n^(n-k) / (n-k)!.

Original entry on oeis.org

1, 2, 15, 136, 1645, 24336, 426979, 8658560, 199234809, 5128019200, 145969492471, 4552809182208, 154404454932325, 5656950010320896, 222655633595044875, 9369696305273798656, 419790650812640438641, 19950175280765680680960, 1002394352017754098219999, 53092232229227200348160000

Offset: 1

Ilya Gutkovskiy, Sep 23 2021

nonn

Cf. A000169, A000312, A001865, A063169, A133297, A134095, A277458, A347994.

Table[n! Sum[(-1)^(k + 1) n^(n - k)/(n - k)!, {k, 1, n}], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-LambertW[-x]/(1 - LambertW[-x]^2), {x, 0, nmax}], x] Range[0, nmax]! // Rest

a(n) = n! * sum(k=1, n, (-1)^(k+1)*n^(n-k)/(n-k)!); \\ Michel Marcus, Sep 23 2021

E.g.f.: -LambertW(-x) / (1 - LambertW(-x)^2).

a(n) = n * A133297(n).

cp's OEIS Frontend

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

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A133297 a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*n^(n-k-1)/(n-k)!.

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A277510 E.g.f.: -1/(1-LambertW(-x))^2.

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A308506 Expansion of e.g.f.: -1/(1-LambertW(-2*x)).

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A332048 a(n) = n! * [x^n] 1 / (1 - LambertW(x))^n.

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A347993 a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * n^(n-k) / (n-k)!.

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A133297 a(n) = n!Sum_{k=1..n} (-1)^(k+1)n^(n-k-1)/(n-k)!.