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 A177533 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up. 11
 1, 1, 2, 6, 24, 120, 719, 5027, 40168, 361080, 3606480, 39623760, 474915803, 6166512899, 86227808578, 1291868401830, 20645144452320, 350547210173280, 6302294420371031, 119600213982762899, 2389140113204434900, 50111866901959213980, 1101140993932295832120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) is the number of permutations of length n that avoid the consecutive pattern 123456 (or equivalently 654321). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 1..30 from Ray Chandler) R. E. L. Aldred, M. D. Atkinson, D. J. McCaughan, Avoiding consecutive patterns in permutations Adv. in Appl. Math., 45(3), 449-461, 2010. A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes Ira M. Gessel, Yan Zhuang, Counting permutations by alternating descents , arXiv:1408.1886 [math.CO], 2014. See displayed equation before Eq. (3), and set m=6. - N. J. A. Sloane, Aug 11 2014 Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018. FORMULA 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 MAPLE 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): seq(a(n), n=0..30);  # Alois P. Heinz, Oct 07 2013 MATHEMATICA 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 *) CROSSREFS Cf. A049774, A117158, A177523. Column k=31 of A242784. Sequence in context: A052398 A047890 A071088 * A122417 A321008 A033644 Adjacent sequences:  A177530 A177531 A177532 * A177534 A177535 A177536 KEYWORD nonn AUTHOR R. H. Hardin, May 10 2010 EXTENSIONS More terms from Ray Chandler, Dec 06 2011 Minor edits by Vaclav Kotesovec, Aug 29 2014 a(0)=1 prepended by Alois P. Heinz, Aug 08 2018 STATUS approved

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Last modified January 17 10:21 EST 2019. Contains 319218 sequences. (Running on oeis4.)