A125500 Expansion of -LambertW(-x^2*exp(x))/x^2.
1, 1, 3, 13, 85, 701, 7261, 89125, 1277865, 20883385, 384194521, 7852225481, 176651705869, 4337650936789, 115468033349397, 3312409332578221, 101881034223806161, 3344745711740899697, 116747433680684736817
Offset: 0
Examples
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! +... - _Paul D. Hanna_, Aug 30 2008
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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GAP
List([0..30],n->Sum([0..n],k->Factorial(n)*(n-k+1)^(k-1)/Factorial(k)*Binomial(k,n-k))); # Muniru A Asiru, Feb 19 2018
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Mathematica
Table[Sum[n!*(n-k+1)^(k-1)/k!*Binomial[k,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 04 2013 *) With[{nmax=30}, CoefficientList[Series[-LambertW[-x^2*Exp[x]]/x^2, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Feb 19 2018 *) -
PARI
{a(n)=local(Ex=exp(x+x*O(x^n)),W=Ex);for(k=0,n,W=exp(x*W)); n!*polcoeff(subst(W,x,x^2*Ex)*Ex,n)} \\ Paul D. Hanna, Jan 02 2007 -
PARI
{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x+x^2*A));n!*polcoeff(A,n)} \\ Paul D. Hanna, Aug 30 2008 -
PARI
{a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(n-k+m)^(k-1)*binomial(k,n-k)))} \\ Paul D. Hanna, Jun 17 2009 -
PARI
{a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(x*(1+x*A)));n!*polcoeff(A^m,n)} \\ Paul D. Hanna, Jun 17 2009 -
PARI
x='x+O('x^30); Vec(serlaplace(-lambertw(-x^2*exp(x))/x^2)) \\ G. C. Greubel, Feb 19 2018
Formula
E.g.f.: A(x) = exp(x + x^2*A(x)). [Paul D. Hanna, Aug 30 2008]
From Paul D. Hanna, Jun 17 2009: (Start)
a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(k,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(k,n-k). (End)
a(n) ~ sqrt((c+1)/2)/(2*c^2) * exp(n*(2*c-1)/2) * n^(n-1), where c = LambertW(exp(-1/2)/2) = 0.2388350311316... - Vaclav Kotesovec, Jan 04 2013
E.g.f.: exp(x - LambertW(-x^2 * exp(x))). - Seiichi Manyama, Apr 20 2023
Extensions
More terms from Paul D. Hanna, Jan 02 2007