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A215882
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Expansion of e.g.f.: -LambertW(-x) / LambertW(x).
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5
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1, 2, 4, 26, 160, 2002, 21184, 395866, 5980160, 149083874, 2933576704, 91549564570, 2222207205376, 83345185392562, 2407376957456384, 105482963294851418, 3534260251308064768, 177194291803516980418, 6757029862401745616896, 381514700506253250858778
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: exp( Sum_{n>=0} 2*(2*n+1)^(2*n) * x^(2*n+1)/(2*n+1)! ).
a(n) = Sum_{k=0..n} -(-1)^k*C(n,k) * (k-1)^(k-1) * (n-k+1)^(n-k-1).
a(n) ~ c * n^(n-1), where c = (1-LambertW(exp(-1))^2)/LambertW(exp(-1)) = 3.31265693390754834... if n is even and c = (1+LambertW(exp(-1))^2)/ LambertW(exp(-1)) = 3.86958601942969593... if n is odd. - Vaclav Kotesovec, Nov 27 2012
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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 26*x^3/3! + 160*x^4/4! + 2002*x^5/5! +... such that A(x) = -LambertW(-x)/LambertW(x) where LambertW(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 625*x^5/5! - 7776*x^6/6! + 117649*x^7/7! - 2097152*x^8/8! +...+ (-n)^(n-1)*x^n/n! +... .
Related expansions: log(A(x)) = 2*x + 18*x^3/3! + 1250*x^5/5! + 235298*x^7/7! + 86093442*x^9/9! +...+ 2*(2*n+1)^(2*n)*x^(2*n+1)/(2*n+1)! +...
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MATHEMATICA
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CoefficientList[Series[-LambertW[-x]/LambertW[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
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PROG
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(PARI) {a(n)=local(LW=sum(m=1, n+1, -(-1)^m*m^(m-1)*x^m/m!)+x^2*O(x^n)); n!*polcoeff(sqrt(-subst(LW, x, -x)/LW), n)}
(PARI) {a(n)=n!*polcoeff(exp(sum(m=0, n, 2*(2*m+1)^(2*m)*x^(2*m+1)/ (2*m+1)!)+x*O(x^n)), n)}
(PARI) {a(n)=sum(k=0, n, -(-1)^k*binomial(n, k)*(k-1)^(k-1)*(n-k+1)^(n-k-1))}
for(n=0, 21, print1(a(n), ", "))
(PARI) x='x+O('x^30); Vec(serlaplace(-lambertw(-x)/lambertw(x))) \\ G. C. Greubel, Feb 19 2018
(GAP) List([0..20], n->Sum([0..n], k->-(-1)^k*Binomial(n, k)*(k-1)^(k-1)*(n-k+1)^(n-k-1))); # Muniru A Asiru, Feb 20 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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