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A187245
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Number of permutations of [n] having no cycle with 2 alternating runs (it is assumed that the smallest element of the cycle is in the first position).
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3
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1, 1, 2, 5, 17, 78, 463, 3315, 27164, 247975, 2492539, 27422698, 328607417, 4266367567, 59686293284, 895068242601, 14320843215019, 243467476610732, 4382635181281015, 83272415871044649, 1665465961530365026, 34974843092354081119, 769445564105823722109
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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E.g.f.: g(z)=exp[(4exp(z)-exp(2z)-3-2z)/4]/(1-z).
a(n) ~ exp(exp(1)-exp(2)/4-5/4) * n! = 0.68455780023755436... * n!. - Vaclav Kotesovec, Mar 15 2014
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EXAMPLE
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a(3)=5 because we have among the 6 permutations of {1,2,3} only 312=(132) has a cycle with 2 alternating runs.
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MAPLE
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g := exp((4*exp(z)-exp(2*z)-3-2*z)*1/4)/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);
# second Maple program:
a:= proc(n) option remember;
`if`(n=0, 1, add(a(n-j)*binomial(n-1, j-1)*
`if`(j=1, 1, (j-1)!-(2^(j-2)-1)), j=1..n))
end:
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MATHEMATICA
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CoefficientList[Series[E^((4*E^x-E^(2*x)-3-2*x)/4)/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Mar 15 2014 *)
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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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