%I #29 Apr 20 2023 09:21:55
%S 1,1,3,13,85,701,7261,89125,1277865,20883385,384194521,7852225481,
%T 176651705869,4337650936789,115468033349397,3312409332578221,
%U 101881034223806161,3344745711740899697,116747433680684736817
%N Expansion of -LambertW(-x^2*exp(x))/x^2.
%H Vincenzo Librandi, <a href="/A125500/b125500.txt">Table of n, a(n) for n = 0..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F E.g.f.: A(x) = exp(x + x^2*A(x)). [_Paul D. Hanna_, Aug 30 2008]
%F From _Paul D. Hanna_, Jun 17 2009: (Start)
%F a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(k,n-k).
%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
%F a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(k,n-k). (End)
%F 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
%F E.g.f.: exp(x - LambertW(-x^2 * exp(x))). - _Seiichi Manyama_, Apr 20 2023
%e 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
%t Table[Sum[n!*(n-k+1)^(k-1)/k!*Binomial[k,n-k],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jan 04 2013 *)
%t 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 *)
%o (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
%o (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
%o (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
%o (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
%o (PARI) x='x+O('x^30); Vec(serlaplace(-lambertw(-x^2*exp(x))/x^2)) \\ _G. C. Greubel_, Feb 19 2018
%o (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
%Y Column k=2 of A362377.
%Y Cf. A143740.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Dec 27 2006
%E More terms from _Paul D. Hanna_, Jan 02 2007