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A371042
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E.g.f. satisfies A(x) = 1 + x^2*exp(x*A(x)).
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3
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1, 0, 2, 6, 12, 140, 1470, 10122, 114296, 1874952, 25462170, 379431470, 7546461252, 151797222876, 3066316693622, 72101615826450, 1843378516587120, 47860832586054032, 1338908395558366386, 40675047500003794902, 1282380661224172506620
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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) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k) * binomial(n-2*k+1,k)/( (n-2*k+1)*(n-2*k)! ).
E.g.f.: 1 - LambertW(-exp(x)*x^3)/x.
a(n) ~ sqrt(1 + LambertW(exp(-1/3)/3)) * n^(n-1) / (exp(n) * 3^(n + 1/2) * LambertW(exp(-1/3)/3)^(n+1)). (End)
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MATHEMATICA
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nmax = 20; CoefficientList[Series[1 - LambertW[-E^x*x^3]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 10 2024 *)
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PROG
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(PARI) a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(n-2*k+1, k)/((n-2*k+1)*(n-2*k)!));
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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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