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A046718 Number of permutations of [ n ] with exactly one 132-pattern and two 123-patterns.

%I A046718 #40 Apr 11 2020 15:22:32
%S A046718 1,4,14,47,152,472,1408,4048,11264,30464,80384,207616,526336,1312768,
%T A046718 3227648,7835648,18808832,44695552,105250816,245825536,569901056,
%U A046718 1312292864,3003121664,6833569792,15468593152,34846277632,78148272128,174533378048,388291887104
%N A046718 Number of permutations of [ n ] with exactly one 132-pattern and two 123-patterns.
%H A046718 Alois P. Heinz, <a href="/A046718/b046718.txt">Table of n, a(n) for n = 4..1000</a>
%H A046718 S. Ekhad, A. Robertson, D. Zeilberger, <a href="http://arxiv.org/abs/math/9903170">The number of permutations with a prescribed number of 132 and 123 patterns</a>, arXiv:math/9903170 [math.CO], 1999.
%H A046718 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F A046718 G.f.: -x^4*(x^3-6*x^2+4*x-1)/(2*x-1)^4.
%F A046718 a(n) = 2^(n-8)*(n^3-11*n^2+54*n-88). - _R. J. Mathar_, Oct 02 2012
%e A046718 a(4) = 1: 1324.
%e A046718 a(5) = 4: 24315, 24351, 41325, 51324.
%e A046718 a(6) = 14: 354216, 354261, 354612, 354621, 435162, 462135, 524316, 524361, 541326, 561324, 624315, 624351, 641325, 651324.
%p A046718 a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-16|32|-24|8>>^(n-4).
%p A046718         <<1, 4, 14, 47>>)[1, 1]:
%p A046718 seq(a(n), n=4..30);  # _Alois P. Heinz_, Oct 01 2012
%t A046718 LinearRecurrence[{8, -24, 32, -16}, {1, 4, 14, 47}, 30] (* _Jean-François Alcover_, Aug 18 2018 *)
%o A046718 (Sage)
%o A046718 def LinearRecurrence4(a0,a1,a2,a3,a4,a5,a6,a7):
%o A046718     x, y, z, u = Integer(a0),Integer(a1),Integer(a2),Integer(a3)
%o A046718     yield x
%o A046718     while True:
%o A046718         x, y, z, u = y, z, u, a7*x+a6*y+a5*z+a4*u
%o A046718         yield x
%o A046718 A046718 = LinearRecurrence4(1, 4, 14, 47, 8, -24, 32, -16)
%o A046718 [next(A046718) for i in range(29)] # _Peter Luschny_, Oct 02 2012
%Y A046718 Cf. A001815, A001787, A011782, A001793.
%K A046718 nonn
%O A046718 4,2
%A A046718 _Len Smiley_
%E A046718 Edited at the suggestion of _R. J. Mathar_ by _Alois P. Heinz_, Oct 01 2012