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A098746 Number of permutations of [1..n] which avoid 4231 and 42513. 6
1, 1, 2, 6, 23, 102, 495, 2549, 13682, 75714, 428882, 2474573, 14492346, 85926361, 514763279, 3111119358, 18946375767, 116147683902, 716179441293, 4438862153246, 27638747494178, 172805469880497, 1084462349973559, 6828717036765622, 43132158190994223, 273204023401012901 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

(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

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

Conjecture: a(n) is the number of permutations of [1..n] which avoid 1342 and 13254. - Alexander Burstein, Oct 19 2017

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Wlodzimierz Bryc, Raouf Fakhfakh, Wojciech Mlotkowski, Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality, arXiv:1708.05343 [math.PR], 2017.

Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016 [Section 2.26].

FORMULA

G.f.: 1 + Sum_{n>=1} ( t^n * Sum_{k=0..n} ((n-l)*binomial(2*k+n, k)/(2*k+n))).

G.f.: sqrt(3)/(sqrt(3)-2*sqrt(x)*sin(asin(3*sqrt(3x)/2)/3)). - Paul Barry, Dec 15 2006

From Gary W. Adamson, Jul 07 2011: (Start)

Let M = the production matrix:

  1, 1;

  1, 2, 1;

  1, 3, 2, 1;

  1, 4, 3, 2, 1;

  1, 5, 4, 3, 2, 1;

  ...

a(n) is the 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). (End)

a(n) ~ 3^(3*n+3/2) / (49 * sqrt(Pi) * 4^n * n^(3/2)). - Vaclav Kotesovec, Mar 17 2014

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

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

Sequence in context: A248900 A120346 A050389 * A245389 A088929 A279573

Adjacent sequences:  A098743 A098744 A098745 * A098747 A098748 A098749

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 30 2004

STATUS

approved

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Last modified February 19 06:46 EST 2018. Contains 299330 sequences. (Running on oeis4.)