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

Author

N. J. A. Sloane, Jun 06 2002

Keywords

• nonn
• easy

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.

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.

Programs

Magma

[1] cat [3*Binomial(2*n,n-1)/(n+2): n in [1..29]]; // Vincenzo Librandi, Jul 12 2017

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 *)

PARI

a(n)=if(n<1,n==0,3*(2*n)!/(n+2)!/(n-1)!)

Sage

[1]+[3*n*catalan_number(n)/(n+2) for n in (1..30)] # G. C. Greubel, Mar 17 2021

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)