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.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Wlodzimierz Bryc, Raouf Fakhfakh, and 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, and Zhicong Lin, Enumeration of inversion sequences avoiding triples of relations, Discrete Applied Mathematics (2019); see also author's copy.
Joanna N. Chen and Zhicong Lin, Combinatorics of the symmetries of ascents in restricted inversion sequences, arXiv:2112.04115 [math.CO], 2021.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Toufik Mansour and Mark Shattuck, Further enumeration results concerning a recent equivalence of restricted inversion sequences, hal-03295362 [math.CO], 2021.
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
G.f: A(x) = 1 + series reversion of x/((1+x)*c(x/(1+x))), where c(x) = (1 - sqrt(1 - 4*x))(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, May 05 2024
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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Oct 30 2004
STATUS
approved