OFFSET
1,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. H. Albert, M. D. Atkinson, Robert Brignall, The enumeration of three pattern classes using monotone grid classes, The Electronic Journal of Combinatorics, vol.19, no.3, (2012).
Darla Kremer and Wai Chee Shiu, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
Index entries for linear recurrences with constant coefficients, signature (18,-147,721,-2369,5505,-9305,11579,-10602,7049,-3304,1032,-192,16).
FORMULA
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
EXAMPLE
There are 22 permutations of length 4 which avoid these two patterns, so a(4) = 22.
MAPLE
A165525 := proc(n)
-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)
+8*A001519(n+2) ;
%/4 ;
end proc:
seq(A165525(n), n=0..20) ; # R. J. Mathar, Aug 19 2022
MATHEMATICA
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 *)
PROG
(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))
/* Joerg Arndt, Aug 16 2012 */
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vincent Vatter, Sep 21 2009
EXTENSIONS
More terms from Joerg Arndt, Aug 16 2012
STATUS
approved