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%I #38 Sep 08 2022 08:45:30
%S 1,1,2,6,16,40,96,224,512,1152,2560,5632,12288,26624,57344,122880,
%T 262144,557056,1179648,2490368,5242880,11010048,23068672,48234496,
%U 100663296,209715200,436207616,905969664,1879048192,3892314112
%N Binomial transform of A124625.
%C Essentially the same as A057711: a(n) = A057711(n) for n >= 1.
%C Number of permutations of length n>=0 avoiding the partially ordered pattern (POP) {1>2, 1>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements. - _Sergey Kitaev_, Dec 08 2020
%H Vincenzo Librandi, <a href="/A129952/b129952.txt">Table of n, a(n) for n = 0..1000</a>
%H Alice L. L. Gao, Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H Alice L. L. Gao, Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4).
%F a(0) = 1, a(1) = 1; for n > 1, a(n) = n*2^(n-2).
%F G.f.: (1-3*x+2*x^2+2*x^3)/(1-2*x)^2.
%F E.g.f.: (1/2)*(x*exp(2*x) + x + 2). - _G. C. Greubel_, Jun 08 2016
%t LinearRecurrence[{4, -4}, {1, 1, 2, 6}, 30] (* _G. C. Greubel_, Jun 08 2016; corrected by _Georg Fischer_, Apr 02 2019 *)
%o (Magma) m:=15; S:=&cat[ [ 1, 2*i ]: i in [0..m] ]; [ &+[ Binomial(j-1, k-1)*S[k]: k in [1..j] ]: j in [1..2*m] ]; // _Klaus Brockhaus_, Jun 17 2007
%o (PARI) {m=29; print1(1, ",", 1, ","); for(n=2, m, print1(n*2^(n-2), ","))} \\ _Klaus Brockhaus_, Jun 17 2007
%Y Cf. A124625, A045623, A057711, A129953 (first differences), A129954 (second differences), A129955 (third differences).
%K nonn,easy
%O 0,3
%A _Paul Curtz_, Jun 10 2007
%E Edited and extended by _Klaus Brockhaus_, Jun 17 2007
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