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A134095
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Expansion of e.g.f. A(x) = 1/(1 - LambertW(-x)^2).
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16
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1, 0, 2, 12, 120, 1480, 22320, 396564, 8118656, 188185680, 4871980800, 139342178140, 4363291266048, 148470651659928, 5455056815237120, 215238256785814500, 9077047768435752960, 407449611073696325536, 19396232794530856894464, 976025303642559490903980
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OFFSET
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0,3
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COMMENTS
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E.g.f. equals the square of the e.g.f. of A060435, where A060435(n) = number of functions f: {1,2,...,n} -> {1,2,...,n} with even cycles only.
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} C(n,k) * (n-k)^k * k^(n-k).
a(n) = (-1)^n*exp(-n)*Integral_{x=-n..oo} x^n*exp(-x) dx. - Thomas Scheuerle, Jan 29 2024
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EXAMPLE
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E.g.f.: A(x) = 1 + 0*x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1480*x^5/5! + ...
The formula A(x) = 1/(1 - LambertW(-x)^2) is illustrated by:
A(x) = 1/(1 - (x + x^2 + 3^2*x^3/3! + 4^3*x^4/4! + 5^4*x^5/5! + ...)^2).
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MAPLE
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seq(simplify(GAMMA(n+1, -n)*(-exp(-1))^n), n=0..20); # Vladeta Jovovic, Oct 17 2007
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MATHEMATICA
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CoefficientList[Series[1/(1-LambertW[-x]^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
a[x0_] := x D[1/x Exp[x], {x, n}] x^n Exp[-x] /. x->x0
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PROG
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(PARI) {a(n)=sum(k=0, n, (n-k)^k*k^(n-k)*binomial(n, k))}
(PARI) /* Generated by e.g.f. 1/(1 - LambertW(-x)^2 ): */
{a(n)=my(LambertW=-x*sum(k=0, n, (-x)^k*(k+1)^(k-1)/k!) +x*O(x^n)); n!*polcoeff(1/(1-subst(LambertW, x, -x)^2), n)}
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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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