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 A177523 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up. 36
 1, 1, 2, 6, 24, 119, 709, 4928, 39144, 349776, 3472811, 37928331, 451891992, 5832672456, 81074690424, 1207441809209, 19181203110129, 323753459184738, 5785975294622694, 109149016813544376, 2167402030585724571, 45190632809497874161, 987099099863360190632 (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 12345 (or equivalently 54321). LINKS Ray Chandler and Alois P. Heinz, Table of n, a(n) for n = 0..400 (terms n = 1..40 from Ray Chandler) 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 , 2014. See displayed equation before Eq. (3), and set m=5. - N. J. A. Sloane, Aug 11 2014 Kaarel Hänni, Asymptotics of descent functions, arXiv:2011.14360 [math.CO], Nov 29 2020, p. 14. Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018. Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020). FORMULA E.g.f.: 1/( Sum_{n>=0} x^(5*n)/(5*n)! - x^(5*n+1)/(5*n+1)! ). a(n)/n! ~ c * (1/r)^n, where r = 1.007187547786015395418998654... is the root of the equation Sum_{n>=0} (r^(5*n)/(5*n)! - r^(5*n+1)/(5*n+1)!) = 0, c = 1.02806793756750152.... - Vaclav Kotesovec, Dec 11 2013 Equivalently, r = 1.00718754778601539541899865400272701484... is the root of the equation (5+sqrt(5)) * cos(sqrt((5-sqrt(5))/2)*r/2) + (5-sqrt(5)) * exp(sqrt(5)*r/2) * cos(sqrt((5+sqrt(5))/2)*r/2) - sqrt(2*(5-sqrt(5))) * sin(sqrt((5-sqrt(5))/2)*r/2) - sqrt(2*(5+sqrt(5))) * exp(sqrt(5)*r/2) * sin(sqrt((5+sqrt(5))/2)*r/2) = 0. - Vaclav Kotesovec, Aug 29 2014 E.g.f.: 10*exp((1+sqrt(5))*x/4) / ((5+sqrt(5)) * cos(sqrt((5-sqrt(5))/2)*x/2) + (5-sqrt(5)) * exp(sqrt(5)*x/2) * cos(sqrt((5+sqrt(5))/2)*x/2) - sqrt(2*(5-sqrt(5))) * sin(sqrt((5-sqrt(5))/2)*x/2) - sqrt(2*(5+sqrt(5))) * exp(sqrt(5)*x/2) * sin(sqrt((5+sqrt(5))/2)*x/2)). - Vaclav Kotesovec, Aug 29 2014 In closed form, c = 5*exp((1+sqrt(5))*r/4) / (r*((5 + sqrt(5)) * cos(sqrt((5 - sqrt(5))/2)*r/2) + (5 - sqrt(5)) * exp(sqrt(5)*r/2) * cos(sqrt((5 + sqrt(5))/2)*r/2))) = 1.0280679375675015201596831656779442465978511664638... . Vaclav Kotesovec, Feb 01 2015 EXAMPLE E.g.f.: A(x) = 1 + x + 2*x^2/2! + 6*x^3/3! + 24*x^4/4! + 119*x^5/5! + 709*x^6/6! +... where A(x) = 1/(1 - x + x^5/5! - x^6/6! + x^10/10! - x^11/11! + x^15/15! - x^16/16! + x^20/20! +...). MATHEMATICA Table[n!*SeriesCoefficient[1/(Sum[x^(5*k)/(5*k)!-x^(5*k+1)/(5*k+1)!, {k, 0, n}]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 11 2013 *) FullSimplify[CoefficientList[Series[10*E^((1+Sqrt[5])*x/4) / ((5+Sqrt[5]) * Cos[Sqrt[(5-Sqrt[5])/2]*x/2] + (5-Sqrt[5]) * E^(Sqrt[5]*x/2) * Cos[Sqrt[(5+Sqrt[5])/2]*x/2] - Sqrt[2*(5-Sqrt[5])] * Sin[Sqrt[(5-Sqrt[5])/2]*x/2] - Sqrt[2*(5+Sqrt[5])] * E^(Sqrt[5]*x/2) * Sin[Sqrt[(5+Sqrt[5])/2]*x/2]), {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Aug 29 2014 *) PROG (PARI) {a(n)=n!*polcoeff(1/sum(m=0, n\5+1, x^(5*m)/(5*m)!-x^(5*m+1)/(5*m+1)!+x^2*O(x^n)), n)} CROSSREFS Cf. A080635, A049774, A117158, A177533. Column k=15 of A242784. Sequence in context: A202233 A202235 A202236 * A005395 A357920 A358611 Adjacent sequences: A177520 A177521 A177522 * A177524 A177525 A177526 KEYWORD nonn AUTHOR R. H. Hardin, May 10 2010 EXTENSIONS More terms from Ray Chandler, Dec 06 2011 a(0)=1 prepended by Alois P. Heinz, Jan 13 2015 STATUS approved

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Last modified June 10 02:46 EDT 2023. Contains 363183 sequences. (Running on oeis4.)