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A254523 Number of permutations of [n] avoiding adjacent step pattern {up}^11.

%I A254523 #11 Feb 01 2015 08:52:35
%S A254523 1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001599,
%T A254523 6227020775,87178290682,1307674357710,20922789683040,355687423926240,
%U A254523 6402373618334400,121645098513933120,2432901965590252800,51090941178938707200,1124000703770606323200
%N A254523 Number of permutations of [n] avoiding adjacent step pattern {up}^11.
%H A254523 Vaclav Kotesovec, <a href="/A254523/b254523.txt">Table of n, a(n) for n = 0..200</a>
%F A254523 E.g.f.: 1 / Sum_{n>=0} (12*n+1-x)*x^(12*n)/(12*n+1)!.
%F A254523 E.g.f.: 6 / (exp(-x) + cos(x) + 2*cos(x/2)*cosh(sqrt(3)*x/2) - cosh(sqrt(3)*x/2)*sin(x/2) - sin(x) + cosh(x/2)*(2*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) - cos(sqrt(3)*x/2)*sinh(x/2) - sqrt(3)*cos(x/2)*sinh(sqrt(3)*x/2)).
%F A254523 a(n)/n! ~ c * (1/r)^n, where r = 1.0000000019270853046730165249753673978954992128247736041276... is the root of the equation Sum_{n>=0} (r^(12*n)/(12*n)! - r^(12*n+1)/(12*n+1)!) = 0, equivalently root of the equation exp(-r) + cos(r) + 2*cos(r/2)*cosh(sqrt(3)*r/2) - cosh(sqrt(3)*r/2)*sin(r/2) - sin(r) + cosh(r/2)*(2*cos(sqrt(3)*r/2) - sqrt(3)*sin(sqrt(3)*r/2)) - cos(sqrt(3)*r/2)*sinh(r/2) - sqrt(3)*cos(r/2)*sinh(sqrt(3)*r/2) = 0, c = 3/(r*sqrt((cosh(sqrt(3)*r/2) * sin(r/2) + sin(r))^2 + 2*sqrt(3)*cosh(r/2) * (cosh(sqrt(3)*r/2) * sin(r/2) + sin(r)) * sin(sqrt(3)*r/2) + 3*cosh(r/2)^2 * sin((sqrt(3)*r)/2)^2)) = 1.0000000210373483515818712802156496756788404534079689145773611990529818919... .
%p A254523 b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p A254523       `if`(t<10, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
%p A254523       add(b(u-j, o+j-1, 0), j=1..u))
%p A254523     end:
%p A254523 a:= n-> b(n, 0, 0):
%p A254523 seq(a(n), n=0..30); # after Alois P. Heinz
%t A254523 CoefficientList[Series[6 / (Exp[-x] + Cos[x] + 2*Cos[x/2] * Cosh[Sqrt[3]*x/2] - Cosh[Sqrt[3]*x/2]*Sin[x/2] - Sin[x] + Cosh[x/2] * (2*Cos[Sqrt[3]*x/2] - Sqrt[3]*Sin[Sqrt[3]*x/2]) - Cos[Sqrt[3]*x/2]*Sinh[x/2] - Sqrt[3]*Cos[x/2]*Sinh[Sqrt[3]*x/2]), {x, 0, 25}], x] * Range[0, 25]!
%Y A254523 Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230232, A230233.
%Y A254523 Column k=2047 of A242784.
%K A254523 nonn
%O A254523 0,3
%A A254523 _Vaclav Kotesovec_, Jan 31 2015