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%I #55 Oct 27 2023 21:38:51
%S 1,1,2,6,24,119,709,4928,39144,349776,3472811,37928331,451891992,
%T 5832672456,81074690424,1207441809209,19181203110129,323753459184738,
%U 5785975294622694,109149016813544376,2167402030585724571,45190632809497874161,987099099863360190632
%N Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up.
%C a(n) is the number of permutations of length n that avoid the consecutive pattern 12345 (or equivalently 54321).
%H Alois P. Heinz, <a href="/A177523/b177523.txt">Table of n, a(n) for n = 0..400</a> (terms n = 1..40 from Ray Chandler)
%H A. Baxter, B. Nakamura, and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/auto.html">Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes</a>
%H Ira M. Gessel, Yan Zhuang, <a href="http://arxiv.org/abs/1408.1886">Counting permutations by alternating descents </a>, 2014. See displayed equation before Eq. (3), and set m=5. - _N. J. A. Sloane_, Aug 11 2014
%H Kaarel Hänni, <a href="https://arxiv.org/abs/2011.14360">Asymptotics of descent functions</a>, arXiv:2011.14360 [math.CO], Nov 29 2020, p. 14.
%H Mingjia Yang, Doron Zeilberger, <a href="https://arxiv.org/abs/1805.06077">Increasing Consecutive Patterns in Words</a>, arXiv:1805.06077 [math.CO], 2018.
%H Mingjia Yang, <a href="https://doi.org/10.7282/t3-d9z1-aw94">An experimental walk in patterns, partitions, and words</a>, Ph. D. Dissertation, Rutgers University (2020).
%F E.g.f.: 1/( Sum_{n>=0} x^(5*n)/(5*n)! - x^(5*n+1)/(5*n+1)! ).
%F 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
%F 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
%F 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
%F 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
%e 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! +...
%e 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! +...).
%t 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 *)
%t 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 *)
%o (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)}
%Y Cf. A080635, A049774, A117158, A177533.
%Y Column k=15 of A242784.
%K nonn
%O 0,3
%A _R. H. Hardin_, May 10 2010
%E More terms from _Ray Chandler_, Dec 06 2011
%E a(0)=1 prepended by _Alois P. Heinz_, Jan 13 2015
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