A071724 a(n) = 3*binomial(2n, n-1)/(n+2), n > 0, with a(0)=1.

1, 1, 3, 9, 28, 90, 297, 1001, 3432, 11934, 41990, 149226, 534888, 1931540, 7020405, 25662825, 94287120, 347993910, 1289624490, 4796857230, 17902146600, 67016296620, 251577050010, 946844533674, 3572042254128, 13505406670700

Offset

0,3

Comments

Number of standard tableaux of shape (n+1,n-1) (n>=1). - Emeric Deutsch, May 30 2004

From Gus Wiseman, Apr 12 2019: (Start)

Also the number of integer partitions (of any positive integer) such that n is the maximum number of unit steps East or South in the Young diagram starting from the upper-left square and ending in a boundary square in the lower-right quadrant. Also the number of integer partitions fitting in a triangular partition of length n but not of length n - 1. For example, the a(0) = 1 through a(4) = 9 partitions are:

() (1) (2) (3)

(11) (22)

(21) (31)

(32)

(111)

(211)

(221)

(311)

(321)

(End)

The sequence (-1)^(n+1)*a(n), for n >= 1 and +1 for n = 0, is the so-called Z-sequence of the Riordan triangle A158909. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. - Wolfdieter Lang, Oct 22 2019

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Joerg Arndt, The a(4)=28 Young tableaux of shape [5, 3].

FindStat, St000384: The maximal part of the shifted composition of an integer partition.

Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.

Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 21.

Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

Gus Wiseman, Young diagrams of all integer partitions fitting in a triangular partition of length n but not of length n - 1, n = 1...4.

Formula

a(n) = A000245(n), n>0.

G.f.: (C(x)-1)*(1-x)/x = (1 + x^2 * C(x)^3)*C(x), where C(x) is g.f. for Catalan numbers, A000108.

G.f.: ((1-sqrt(1-4*x))/(2*x)-1)*(1-x)/x = A(x) satisfies x^2*A(x)^2 + (x-1)*(2*x-1)*A(x) + (x-1)^2 = 0.

G.f.: 1 + x*C(x)^3, where C(x) is g.f. for the Catalan numbers (A000108). Sequence without the first term is the 3-fold convolution of the Catalan sequence. - Emeric Deutsch, May 30 2004

a(n) is the n-th moment of the function defined on the segment (0, 4) of x axis: a(n) = Integral_{x=0..4} x^n*(-x^(1/2)*cos(3*arcsin((1/2)*x^(1/2)))/Pi) dx, n=0, 1... . - Karol A. Penson, Sep 29 2004

D-finite with recurrence -(n+2)*(n-1)*a(n) + 2*n*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Jul 10 2017

a(n) ~ c*2^(2*n)*n^(-3/2), where c = 3/sqrt(Pi). - Stefano Spezia, Sep 23 2022

From Amiram Eldar, Sep 29 2022: (Start)

Sum_{n>=0} 1/a(n) = 14*(Pi/(3*sqrt(3)) + 1)/9.

Sum_{n>=0} (-1)^n/a(n) = 18/25 - 164*log(phi)/(75*sqrt(5)), where phi is the golden ratio (A001622). (End)

Maple

A071724:= n-> 3*binomial(2*n, n-1)/(n+2); 1, seq(A071724(n), n=1..30); # G. C. Greubel, Mar 17 2021

Mathematica

Join[{1}, Table[3Binomial[2n, n-1]/(n+2), {n, 1, 30}]] (* Vincenzo Librandi, Jul 12 2017 *)
nn=7;
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
allip=Join@@Table[IntegerPartitions[n], {n, 0, nn*(nn+1)/2}];
Table[Length[Select[allip, otbmax[#]==n&]], {n, 0, nn}] (* Gus Wiseman, Apr 12 2019 *)

Prog

(PARI) a(n)=if(n<1, n==0, 3*(2*n)!/(n+2)!/(n-1)!)
(Magma) [1] cat [3*Binomial(2*n, n-1)/(n+2): n in [1..29]]; // Vincenzo Librandi, Jul 12 2017
(Sage) [1]+[3*n*catalan_number(n)/(n+2) for n in (1..30)] # G. C. Greubel, Mar 17 2021

Crossrefs

Cf. A000245, A001622, A002421, A030237, A051924, A096771, A115720, A158909.

Cf. A325169, A325188, A325193, A325195, A325200.

Number of times n appears in A065770.

Column sums of A325189.

Row sums of A030237.

Sequence in context: A094826 A033190 A370074 * A000245 A143739 A189940

Adjacent sequences: A071721 A071722 A071723 * A071725 A071726 A071727

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 06 2002

Status

approved