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A177533
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Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up.
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11
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1, 1, 2, 6, 24, 120, 719, 5027, 40168, 361080, 3606480, 39623760, 474915803, 6166512899, 86227808578, 1291868401830, 20645144452320, 350547210173280, 6302294420371031, 119600213982762899, 2389140113204434900, 50111866901959213980, 1101140993932295832120
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of permutations of length n that avoid the consecutive pattern 123456 (or equivalently 654321).
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LINKS
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FORMULA
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a(n)/n! ~ 1.005827831279392186... * (1/r)^n, where r = 1.0011988273240623031887... is the root of the equation Sum_{n>=0} (r^(6*n)/(6*n)! - r^(6*n+1)/(6*n+1)!) = 0. - Vaclav Kotesovec, Dec 11 2013
Equivalently, a(n)/n! ~ c * (1/r)^n, where r = 1.00119882732406230318870210972855430833421618931012450844128... is the root of the equation 2 + exp(r/2) * (3 + exp(r)) * cos(sqrt(3)*r/2) = 2 * sqrt(3) * exp(r) * cosh(r/2) * sin(sqrt(3)*r/2), c = sqrt(3) / (2 * r * cosh(r/2) * sin(sqrt(3)*r/2)) = 1.0058278312793921866941324506580803251270892126827302878865925027445... . - Vaclav Kotesovec, Aug 23 2014
E.g.f. (Aldred, Atkinson, McCaughan, 2010): 3/(exp(x/2) * cos(x*sqrt(3)/2+Pi/3) + sqrt(3) * exp(-x/2) * cos(x*sqrt(3)/2+Pi/6) + exp(-x)). - Vaclav Kotesovec, Aug 23 2014
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MAPLE
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
`if`(t<4, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
add(b(u-j, o+j-1, 0), j=1..u))
end:
a:= n-> b(n, 0, 0):
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MATHEMATICA
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Table[n!*SeriesCoefficient[1/(Sum[x^(6*k)/(6*k)!-x^(6*k+1)/(6*k+1)!, {k, 0, n}]), {x, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Dec 11 2013 *)
Rest[CoefficientList[Series[3/(E^(x/2) * Cos[x*Sqrt[3]/2+Pi/3] + Sqrt[3] * E^(-x/2) * Cos[x*Sqrt[3]/2+Pi/6] + E^(-x)), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Aug 23 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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EXTENSIONS
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STATUS
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approved
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