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A125500 Expansion of -LambertW(-x^2*exp(x))/x^2. 5
1, 1, 3, 13, 85, 701, 7261, 89125, 1277865, 20883385, 384194521, 7852225481, 176651705869, 4337650936789, 115468033349397, 3312409332578221, 101881034223806161, 3344745711740899697, 116747433680684736817 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

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

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

Sequence in context: A225236 A152789 A192943 * A121679 A246387 A023037

Adjacent sequences:  A125497 A125498 A125499 * A125501 A125502 A125503

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Dec 27 2006

EXTENSIONS

More terms from Paul D. Hanna, Jan 02 2007

STATUS

approved

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Last modified October 15 21:06 EDT 2018. Contains 316237 sequences. (Running on oeis4.)