OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Lambert W-Function.
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
EXAMPLE
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
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Dec 27 2006
EXTENSIONS
More terms from Paul D. Hanna, Jan 02 2007
STATUS
approved