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%I #27 Apr 08 2021 12:43:30
%S 1,1,2,4,11,39,161,784,4368,27260,189540,1448860,12076408,109102564,
%T 1061259548,11060323280,122963473024,1452414435968,18164949751872,
%U 239807221886128,3332441297971360,48624372236312912,743273838888233264,11878134680411900928
%N The number of permutations avoiding simultaneously consecutive patterns 123 and 231.
%C Terms a(11) through a(14) calculated by Elizalde and Noy, who state that an involved explicit form for the e.g.f. can be found in terms of integrals containing the error function.
%D S. Elizalde and M. Noy, Consecutive patterns in permutations (Theorem 5.1), Adv. Appl. Math. 30 (2003) 110-125.
%H Alois P. Heinz, <a href="/A173938/b173938.txt">Table of n, a(n) for n = 0..200</a>
%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 S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns, Ars Combinatoria 76 (2005), 321-350 [<a href="http://www.ams.org/mathscinet-getitem?mr=2152770">MR2152770</a>]
%H S. Kitaev and T. Mansour, <a href="https://web.archive.org/web/20061013131319/http://www.ru.is/kennarar/sergey/index_files/Papers/multi_avoid_gen_patterns.pdf">On multi-avoidance of generalized patterns</a>
%F For all n >= 3, A(n) = a(n-1) + a(n;1) + a(n;2) + ... + a(n;n-1), where for all 1<= i <= n, a(n;i)= Sum_{j=1..i-1} a(n-1;j) + Sum_{j=i..n-2} (n-1-j)*a*(n-2;j), and a(3;1)=1, a(3;2)=1 a(3;3)=2.
%F a(n) ~ c * d^n * n!, where d = A246041 = 0.6948193008667305362671927506... is the root of the equation sqrt(2*Pi)*(erfi(1/sqrt(2)) + erfi((1/d-1)/sqrt(2))) = 2*exp(1/2), c = 1.991594102047693697258367189... . - _Vaclav Kotesovec_, Aug 23 2014
%e Example: For n = 3 a(3) = 4 since 132, 213, 312, and 321 are the 3-permutations avoiding 123 and 231.
%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
%p add(b(u-j, o+j-1, 0), j=1..`if`(t>0, min(u, t-1), u))+
%p `if`(t>0, 0, add(b(u+j-1, o-j, j), j=1..o)))
%p end:
%p a:= n-> b(n, 0, 0):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 25 2013
%t FullSimplify[CoefficientList[Series[1 + Integrate[(2*Sqrt[E]/(2*Sqrt[E] - Sqrt[2*Pi]*Erfi[1/Sqrt[2]] - Sqrt[2*Pi] * Erfi[(-1+x)/Sqrt[2]]))*((E^(1/2*(-1+x)^2) * (2 + Sqrt[2*E*Pi]*Erf[1/Sqrt[2]] - Pi*Erf[1/Sqrt[2]]*Erfi[1/Sqrt[2]] + Erf[(-1+x)/Sqrt[2]]*(Sqrt[2*E*Pi] - Pi*Erfi[1/Sqrt[2]]) + HypergeometricPFQ[{1, 1}, {3/2, 2}, -1/2] - (-1+x)^2 * HypergeometricPFQ[{1, 1}, {3/2, 2}, -1/2*(-1+x)^2])) / (2*Sqrt[E] - Sqrt[2*Pi]*(Erfi[1/Sqrt[2]] + Erfi[(-1+x)/Sqrt[2]]))), x], {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Aug 22 2014 *)
%K nonn
%O 0,3
%A Signy Olafsdottir (signy06(AT)ru.is), Mar 03 2010
%E a(15)-a(23) from _Alois P. Heinz_, Oct 25 2013
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