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A165525
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Number of permutations of length n which avoid the patterns 4321 and 2143.
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1
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1, 2, 6, 22, 86, 333, 1235, 4339, 14443, 45770, 138988, 407134, 1157576, 3212157, 8740499, 23416169, 61973483, 162492830, 423077186, 1095978346, 2829227612, 7287399119, 18748151799, 48213813401, 124015241701, 319200003734, 822375826202, 2121215940232, 5478421803252, 14167748769035, 36687397002789, 95122416842861
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OFFSET
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1,2
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (18,-147,721,-2369,5505,-9305,11579,-10602,7049,-3304,1032,-192,16).
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FORMULA
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G.f.: x*p/((1-2*x)^4*(1-x)^7*(1-3*x+x^2)) where p = 1 - 16*x + 117*x^2 - 513*x^3 + 1499*x^4 - 3074*x^5 + 4530*x^6 - 4827*x^7 + 3691*x^8 - 1968*x^9 + 690*x^10 - 150*x^11 + 16*x^12. - Joerg Arndt, Aug 16 2012
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EXAMPLE
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There are 22 permutations of length 4 which avoid these two patterns, so a(4) = 22.
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MAPLE
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-84-29*n+n^3+263/90*n^2+1/45*n^6+19/18*n^4
+2^n*(n^2/2-110*n/3+72+n^3/6)
%/4 ;
end proc:
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MATHEMATICA
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CoefficientList[Series[(1 -16*x +117*x^2 -513*x^3 +1499*x^4 -3074*x^5 +4530*x^6 -4827*x^7 +3691*x^8 -1968*x^9 +690*x^10 -150*x^11 +16*x^12) / ((1-2*x)^4*(1-x)^7*(1-3*x+x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 18 2012 *)
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PROG
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(PARI)
p=1-16*x+117*x^2-513*x^3+1499*x^4-3074*x^5+4530*x^6 -4827*x^7 +3691*x^8 -1968*x^9+690*x^10-150*x^11+16*x^12;
/* coefficient of x^5 is given as 3064 in Albert et al. reference */
gf=x*p/((1-2*x)^4*(1-x)^7*(1-3*x+x^2));
vA165525=Vec(gf + O('x^33))
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 -16*x +117*x^2 -513*x^3 +1499*x^4 -3074*x^5 +4530*x^6 -4827*x^7 +3691*x^8 -1968*x^9 +690*x^10 -150*x^11 +16*x^12) / ((1-2*x)^4*(1-x)^7*( 1-3*x+x^2)))); // G. C. Greubel, Oct 21 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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