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A071724 a(n) = 3*C(2n,n-1)/(n+2), n>0. a(0)=1. 14
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 (list; graph; refs; listen; history; text; internal format)
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)

LINKS

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

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

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

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

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

G.f.: 1+xC(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)= int(x^n*(-x^(1/2)*cos(3*arcsin(1/2*x^(1/2)))/Pi), x=0..4), n=0, 1... . - Karol A. Penson, Sep 29 2004

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

MATHEMATICA

Join[{1}, Table[3 Binomial[2 n, 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

CROSSREFS

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

Cf. A002421.

Number of times n appears in A065770.

Column sums of A325189.

Cf. A051924, A096771, A115720, A325169, A325188, A325193, A325195, A325200.

Sequence in context: A094803 A094826 A033190 * A000245 A143739 A189940

Adjacent sequences:  A071721 A071722 A071723 * A071725 A071726 A071727

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 06 2002

STATUS

approved

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Last modified May 21 10:48 EDT 2019. Contains 323443 sequences. (Running on oeis4.)