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%I #69 Mar 03 2024 14:44:04
%S 1,1,3,9,28,90,297,1001,3432,11934,41990,149226,534888,1931540,
%T 7020405,25662825,94287120,347993910,1289624490,4796857230,
%U 17902146600,67016296620,251577050010,946844533674,3572042254128,13505406670700
%N a(n) = 3*binomial(2n, n-1)/(n+2), n > 0, with a(0)=1.
%C Number of standard tableaux of shape (n+1,n-1) (n>=1). - _Emeric Deutsch_, May 30 2004
%C From _Gus Wiseman_, Apr 12 2019: (Start)
%C 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:
%C () (1) (2) (3)
%C (11) (22)
%C (21) (31)
%C (32)
%C (111)
%C (211)
%C (221)
%C (311)
%C (321)
%C (End)
%C 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
%H Vincenzo Librandi, <a href="/A071724/b071724.txt">Table of n, a(n) for n = 0..1000</a>
%H Joerg Arndt, <a href="/A071724/a071724.txt">The a(4)=28 Young tableaux of shape [5, 3]</a>.
%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000384">St000384: The maximal part of the shifted composition of an integer partition</a>.
%H Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, <a href="https://arxiv.org/abs/2301.10830">Counting Parking Sequences and Parking Assortments Through Permutations</a>, arXiv:2301.10830 [math.CO], 2023.
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 21.
%H Zhicong Lin, David G.L. Wang, and Tongyuan Zhao, <a href="https://arxiv.org/abs/2103.04599">A decomposition of ballot permutations, pattern avoidance and Gessel walks</a>, arXiv:2103.04599 [math.CO], 2021.
%H Murray Tannock, <a href="http://hdl.handle.net/1946/25589">Equivalence classes of mesh patterns with a dominating pattern</a>, MSc Thesis, Reykjavik Univ., May 2016.
%H Gus Wiseman, <a href="/A071724/a071724.png">Young diagrams of all integer partitions fitting in a triangular partition of length n but not of length n - 1, n = 1...4.</a>
%F a(n) = A000245(n), n>0.
%F 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.
%F 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.
%F 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
%F 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
%F 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
%F a(n) ~ c*2^(2*n)*n^(-3/2), where c = 3/sqrt(Pi). - _Stefano Spezia_, Sep 23 2022
%F From _Amiram Eldar_, Sep 29 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 14*(Pi/(3*sqrt(3)) + 1)/9.
%F Sum_{n>=0} (-1)^n/a(n) = 18/25 - 164*log(phi)/(75*sqrt(5)), where phi is the golden ratio (A001622). (End)
%p A071724:= n-> 3*binomial(2*n, n-1)/(n+2); 1,seq(A071724(n), n=1..30); # _G. C. Greubel_, Mar 17 2021
%t Join[{1}, Table[3Binomial[2n, n-1]/(n+2), {n,1,30}]] (* _Vincenzo Librandi_, Jul 12 2017 *)
%t nn=7;
%t otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t allip=Join@@Table[IntegerPartitions[n],{n,0,nn*(nn+1)/2}];
%t Table[Length[Select[allip,otbmax[#]==n&]],{n,0,nn}] (* _Gus Wiseman_, Apr 12 2019 *)
%o (PARI) a(n)=if(n<1,n==0,3*(2*n)!/(n+2)!/(n-1)!)
%o (Magma) [1] cat [3*Binomial(2*n,n-1)/(n+2): n in [1..29]]; // _Vincenzo Librandi_, Jul 12 2017
%o (Sage) [1]+[3*n*catalan_number(n)/(n+2) for n in (1..30)] # _G. C. Greubel_, Mar 17 2021
%Y Cf. A000245, A001622, A002421, A030237, A051924, A096771, A115720, A158909.
%Y Cf. A325169, A325188, A325193, A325195, A325200.
%Y Number of times n appears in A065770.
%Y Column sums of A325189.
%Y Row sums of A030237.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Jun 06 2002
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