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A125500
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Expansion of -LambertW(-x^2*exp(x))/x^2.
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20
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1, 1, 3, 13, 85, 701, 7261, 89125, 1277865, 20883385, 384194521, 7852225481, 176651705869, 4337650936789, 115468033349397, 3312409332578221, 101881034223806161, 3344745711740899697, 116747433680684736817
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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