A376578 E.g.f. A(x) satisfies A(x) = exp(x*(1 + x^3)*A(x)).
1, 1, 3, 16, 149, 1656, 22567, 367144, 7020729, 153727408, 3792553451, 104109493104, 3149158146037, 104087591605336, 3732676321011471, 144352230854996056, 5988802115089340273, 265333292818927985376, 12503715930615717645139, 624511093178389790592352
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Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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PARI
a(n) = n!*sum(k=0, n\4, (n-3*k+1)^(n-3*k-1)*binomial(n-3*k, k)/(n-3*k)!);
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x^3)))))
Formula
a(n) = n! * Sum_{k=0..floor(n/4)} (n-3*k+1)^(n-3*k-1) * binomial(n-3*k,k)/(n-3*k)!.
E.g.f.: exp( -LambertW(-x*(1+x^3)) ).
From Vaclav Kotesovec, Oct 10 2024: (Start)
E.g.f.: -LambertW(-x*(1+x^3))/(x*(1+x^3)).
a(n) ~ sqrt(1 + 4*r^3) * n^(n-1) / (exp(n - 3/2) * r^(n - 1/2)), where r = 0.3524487947753533587882657130849651130602232209406... is the positive real root of the equation r*(1 + r^3) = exp(-1). (End)