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%I #21 Aug 15 2018 18:49:03
%S 1,1,2,6,24,120,720,5040,40320,362880,3628799,39916779,479001228,
%T 6227014404,87178179816,1307672369640,20922752672640,355686706327680,
%U 6402359109968640,121644792614741760,2432895242801771955,51090787299486057355,1123997039003038423610
%N Number of permutations of [n] avoiding adjacent step pattern {up}^9.
%H Alois P. Heinz, <a href="/A230232/b230232.txt">Table of n, a(n) for n = 0..450</a>
%F E.g.f.: 1 / Sum_{n>=0} (10*n+1-x)*x^(10*n)/(10*n+1)!.
%F a(n)/n! ~ c * (1/r)^n, where r = 1.0000002505217051890946793081039639693008257169189079028339632923816... is the root of the equation Sum_{n>=0} (r^(10*n)/(10*n)! - r^(10*n+1)/(10*n+1)!) = 0, c = 1.000002229648140602899529055054469878816530201510267349345270187155... . - _Vaclav Kotesovec_, Jan 17 2015
%F E.g.f.: 10 / (2/exp(x) - sqrt(2*(5 - sqrt(5))) * cosh((1/4)*(1 + sqrt(5))*x) * sin(sqrt((1/8)*(5 - sqrt(5)))*x) - sqrt(2*(5 + sqrt(5))) * cosh((1/4)*(sqrt(5) - 1)*x) * sin(sqrt((1/8)*(5 + sqrt(5)))*x) + cos(sqrt((1/8)*(5 + sqrt(5)))*x) * (4*cosh((1/4)*(sqrt(5) - 1)*x) - (sqrt(5) - 1)*sinh((1/4)*(sqrt(5) - 1)*x)) - cos(sqrt((1/8)*(5 - sqrt(5)))*x) * ((1 + sqrt(5))*sinh((1/4)*(1 + sqrt(5))*x) - 4*cosh((1/4)*(1 + sqrt(5))*x))). - _Vaclav Kotesovec_, Jan 31 2015
%F In closed form, c = 5 / (r * (sqrt(10 - 2*sqrt(5)) * cosh((sqrt(5)+1)*r/4) * sin(sqrt((5 - sqrt(5))/2)*r/2) + sqrt(2*(5 + sqrt(5))) * cosh((sqrt(5)-1)*r/4) * sin(sqrt((5 + sqrt(5))/2)*r/2))). - _Vaclav Kotesovec_, Feb 01 2015
%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p `if`(t<8, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
%p add(b(u-j, o+j-1, 0), j=1..u))
%p end:
%p a:= n-> b(n, 0, 0):
%p seq(a(n), n=0..30);
%t nn=20;r=9;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* _Geoffrey Critzer_, Feb 25 2014 *)
%t FullSimplify[CoefficientList[Series[10/(2/E^x - Sqrt[2*(5 - Sqrt[5])]* Cosh[(1/4)*(1 + Sqrt[5])*x]* Sin[Sqrt[(1/8)*(5 - Sqrt[5])]*x] - Sqrt[2*(5 + Sqrt[5])]*Cosh[(1/4)*(Sqrt[5] - 1)* x]*Sin[Sqrt[(1/8)*(5 + Sqrt[5])]*x] + Cos[Sqrt[(1/8)*(5 + Sqrt[5])]*x]* (4*Cosh[(1/4)*(Sqrt[5] - 1)*x] - (Sqrt[5] - 1)*Sinh[(1/4)*(Sqrt[5] - 1)*x]) - Cos[Sqrt[(1/8)*(5 - Sqrt[5])]*x]* ((1 + Sqrt[5])*Sinh[(1/4)*(1 + Sqrt[5])*x] - 4*Cosh[(1/4)*(1 + Sqrt[5])*x])), {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 31 2015 *)
%Y Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230233.
%Y Column k=511 of A242784.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Oct 12 2013
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