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A098746 Number of permutations of [1..n] which avoid 4231 and 42513. 5

%I

%S 1,1,2,6,23,102,495,2549,13682,75714,428882,2474573,14492346,85926361,

%T 514763279,3111119358,18946375767,116147683902,716179441293,

%U 4438862153246,27638747494178,172805469880497,1084462349973559,6828717036765622,43132158190994223,273204023401012901

%N Number of permutations of [1..n] which avoid 4231 and 42513.

%C (a(n))_{n>=1} is the INVERT transform of (u(n))_{n>=1}:=(1,1,3,12,55,273,...), the ternary numbers A001764. - David Callan Nov 21 2011

%C a(n) = number of Dyck paths of semilength 2n for which all descents are of even length (counted by A001764) with no valley vertices at height 1. For example, a(2)=2 counts UUUUDDDD, UUDDUUDD. - David Callan, Nov 21 2011

%D M. H. Albert et al., Restricted permutations and queue jumping, Discrete Math., 287 (2004), 129-133.

%F G.f.: 1+Sum( t^n * Sum( (n-l)*binomial(2*l+n, l)/(2*l+n), l=0..n ), n=1..oo).

%F G.f.: sqrt(3)/(sqrt(3)-2*sqrt(x)*sin(asin(3*sqrt(3x)/2)/3)); - _Paul Barry_, Dec 15 2006

%F Let M = the production matrix:

%F 1, 1

%F 1, 2, 1

%F 1, 3, 2, 1

%F 1, 4, 3, 2, 1

%F 1, 5, 4, 3, 2, 1

%F ...

%F a(n) = upper left term in M^n, with sum of top row terms = a(n+1). Example: top row of M^3 = (6, 11, 5, 1), where a(3) = 6 and a(4) = 23 = (6 + 11 + 5 + 1). - Gary W. Adamson, Jul 07 2011

%F a(n) ~ 3^(3*n+3/2) / (49 * sqrt(Pi) * 4^n * n^(3/2)). - _Vaclav Kotesovec_, Mar 17 2014

%F Conjecture: 2*(2*n-1)*(n-1)*a(n) +3*(11*n^2-67*n+76)*a(n-1) +3*(-155*n^2+931*n-1356)*a(n-2) +(469*n^2-2799*n+4070)*a(n-3) -48*(3*n-8)*(3*n-10)*a(n-4)=0. - _R. J. Mathar_, May 30 2014

%p 1+add( t^n * add( (n-l)*binomial(2*l+n,l)/(2*l+n), l=0..n ), n=1..30);

%t Flatten[{1,Table[Sum[(n-j)*Binomial[2*j+n,j]/(2*j+n),{j,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Mar 17 2014 *)

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Oct 30 2004

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Last modified December 19 04:51 EST 2014. Contains 252175 sequences.