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%I #23 Jul 06 2024 08:46:49
%S 1,3,11,45,198,919,4446,22239,114347,601722,3229614,17632437,97707195,
%T 548538588,3115293151,17875151109,103511938302,604392787819,
%U 3555410248782,21057224371290,125484804821226,752020468811244,4530163818778839,27419805899781843,166694596163875206
%N Number of permutations of length n avoiding the pattern 1324 and with 1 appearing before n.
%C This sequence counts the number of permutations of size n written in one-line notation that avoid the pattern 1324 and have the 1 appearing before the n.
%H Juan B. Gil, Oscar A. Lopez, and Michael D. Weiner, <a href="http://arxiv.org/abs/2311.18227">A positional statistic for 1324-avoiding permutations</a>, arXiv:2311.18227 [math.CO], 2024.
%F G.f.: A(x) = (x*(B(x)-2))/(3-B(x)), where B(x) is the g.f. for A000139. (See arxiv paper by Gil, Lopez, Weiner.)
%F G.f. satisfies 0 = x^4*(8*x-1) + x^2*(9*x-1)*(4*x-1)*A(x) + x*(6*x-1)*(9*x-2)*A(x)^2 + (27*x^2-9*x+1)*A(x)^3.
%F a(n) ~ 3^(3*n - 9/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n - 5)). - _Vaclav Kotesovec_, Jul 06 2024
%e For n=4, a(4)=11 is counting the permutations (in one-line notation): 1234, 1243, 1342, 1423, 1432, 2134, 2143, 2314, 3124, 3142, 3214.
%p f:= proc(n) f(n):= 2*(3*n)!/((2*n+1)!*(n+1)!) end:
%p a:= proc(n) option remember; `if`(n=1, 1,
%p add(a(n-i)*f(i), i=1..n))
%p end:
%p seq(a(n), n=2..26); # _Alois P. Heinz_, Apr 26 2024
%t a[1] = 1; a[n_] := a[n] = 2*Sum[a[n-k]*(3*k)!/((2*k + 1)!*(k+1)!), {k, 1, n-1}]; Table[a[n], {n, 2, 30}] (* _Vaclav Kotesovec_, Jul 06 2024 *)
%Y Invert of shifted A000139.
%K nonn,changed
%O 2,2
%A _Michael D. Weiner_, Apr 26 2024
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