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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.

Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.

Wlodzimierz Bryc, Raouf Fakhfakh, Wojciech Mlotkowski, Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality, arXiv:1708.05343 [math.PR], 2017-2019. Also in Probability and Mathematical Statistics (2019), Vol. 39, No. 2, 237-258.

Wenqin Cao, Emma Yu Jin, Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019).

Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

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-k)*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 *)

PROG

(PARI) a(n) = {my(k = 1); if(n > 0, k = sum(j = 0, n, (n-j)*binomial(2*j+n, j)/(2*j+n))); k; } \\ Jinyuan Wang, Aug 03 2019

CROSSREFS

Sequence in context: A248900 A120346 A050389 * A245389 A088929 A279573

Adjacent sequences:  A098743 A098744 A098745 * A098747 A098748 A098749

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane, Oct 30 2004

STATUS

approved

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Last modified March 29 21:32 EDT 2020. Contains 333117 sequences. (Running on oeis4.)