login
Expansion of x*(1-6*x+10*x^2)/(1-4*x+2*x^2)^2.
1

%I #27 Feb 18 2024 15:19:46

%S 0,1,2,6,24,100,408,1624,6336,24336,92320,346720,1291392,4776512,

%T 17562496,64245120,233969664,848748800,3068269056,11057710592,

%U 39740405760,142466343936,509572929536,1818872207360,6480018948096

%N Expansion of x*(1-6*x+10*x^2)/(1-4*x+2*x^2)^2.

%C Binomial transform of A093969 (n*Pell(n-2)).

%C For n>0, a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 4>2, 4>3} of length 5. That is, the number of length n permutations having no subsequences of length 5 in which the elements in positions 1 and 4 are larger than the elements in positions 2 and 3. - _Sergey Kitaev_, Dec 11 2020

%H Michael De Vlieger, <a href="/A094012/b094012.txt">Table of n, a(n) for n = 0..1871</a>

%H Alice L. L. Gao and 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 and 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_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-20,16,-4).

%F a(n) = (n/4)*( (3+2*sqrt(2))*(2-sqrt(2))^n + (3-2*sqrt(2))*(2+sqrt(2))^n ).

%F From _G. C. Greubel_, Dec 27 2021: (Start)

%F a(2*n) = n*2^(n-1)*A002203(2*n-2).

%F a(2*n+1) = (2*n+1)*2^n*A000129(2*n-2). (End)

%e G.f. = x + 2*x^2 + 6*x^3 + 24*x^4 + 100*x^5 + 408*x^6 + ... - _Michael Somos_, Dec 30 2021

%t LinearRecurrence[{8,-20,16,-4}, {0,1,2,6}, 25] (* _Michael De Vlieger_, May 01 2019 *)

%t a[ n_] := n 2^(Quotient[n-3, 2] + Mod[n, 2]) If[OddQ[n], Fibonacci, LucasL][n-2, 2]; (* _Michael Somos_, Dec 30 2021 *)

%t CoefficientList[Series[x (1-6x+10x^2)/(1-4x+2x^2)^2,{x,0,30}],x] (* _Harvey P. Dale_, Feb 18 2024 *)

%o (Magma) I:=[0,1,2,6]; [n le 4 select I[n] else 8*Self(n-1) - 20*Self(n-2) +16*Self(n-3) -4*Self(n-4): n in [1..30]]; // _G. C. Greubel_, Dec 27 2021

%o (Sage)

%o def a(n):

%o if (n%2==0): return (n/2)*2^(n/2 -1)*lucas_number2(n-2,2,-1)

%o else: return n*2^((n-1)/2)*lucas_number1(n-2,2,-1)

%o [a(n) for n in range(30)] # _G. C. Greubel_, Dec 27 2021

%Y Cf. A000129, A002203, A093969.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Apr 21 2004