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A277221
Number of permutations of length n which avoid the patterns 4123, 1324, and 3124.
2
1, 1, 2, 6, 21, 78, 297, 1143, 4419, 17119, 66386, 257621, 1000407, 3887666, 15119991, 58856167, 229312425, 894263633, 3490636794, 13637575699, 53327459013, 208703945330, 817447047177, 3204204114421, 12568821046236, 49336156718513, 193783005926727, 761604774463568
OFFSET
0,3
COMMENTS
a(n) is also the number of permutations of length n which avoid the patterns 4123, 1324, and 1423.
LINKS
D. Callan, T. Mansour, Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns, arXiv:1705.00933 [math.CO] (2017), Table 1 No 231.
Sam Miner, Enumeration of several two-by-four classes, arXiv:1610.01908 [math.CO], 2016.
FORMULA
G.f.: (1 - 3x) * (1 + sqrt(1 - 4x)) / (2 * (1 - 3x + x^2) * sqrt(1 - 4x)).
a(n) ~ 2^(2*n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 07 2016
n*(n-3)*a(n) +(-7*n^2+23*n-12)*a(n-1) +(13*n^2-45*n+36)*a(n-2) -2*(2*n-3)*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 09 2017
MATHEMATICA
CoefficientList[Series[(1-3*x)*(1+Sqrt[1-4*x])/(2*(1-3*x+x^2)*Sqrt[1- 4*x]), {x, 0, 50}], x] (* G. C. Greubel, Apr 09 2017 *)
PROG
(PARI) x='x+O('x^44); Vec((1 - 3*x) * (1 + sqrt(1 - 4*x)) / (2 * (1 - 3*x + x^2) * sqrt(1 - 4*x))) \\ Joerg Arndt, Oct 06 2016
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1 -3*x)*(1+Sqrt(1-4*x))/(2*(1-3*x+x^2)*Sqrt(1-4*x)))); // G. C. Greubel, Oct 22 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Sam Miner, Oct 05 2016
STATUS
approved