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A051924 a(n) = binomial(2*n,n) - binomial(2*n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

%I A051924 #171 May 23 2025 00:15:02
%S A051924 1,4,14,50,182,672,2508,9438,35750,136136,520676,1998724,7696444,
%T A051924 29716000,115000920,445962870,1732525830,6741529080,26270128500,
%U A051924 102501265020,400411345620,1565841089280,6129331763880,24014172955500,94163002754652,369507926510352
%N A051924 a(n) = binomial(2*n,n) - binomial(2*n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).
%C A051924 Number of partitions with Ferrers plots that fit inside an n X n box, but not in an n-1 X n-1 box. - _Wouter Meeussen_, Dec 10 2001
%C A051924 From _Benoit Cloitre_, Jan 29 2002: (Start)
%C A051924 Let m(1,j)=j, m(i,1)=i and m(i,j) = m(i-1,j) + m(i,j-1); then a(n) = m(n,n):
%C A051924   1  2  3  4 ...
%C A051924   2  4  7 11 ...
%C A051924   3  7 14 25 ...
%C A051924   4 11 25 50 ... (End)
%C A051924 This sequence also gives the number of clusters and non-crossing partitions of type D_n. - _F. Chapoton_, Jan 31 2005
%C A051924 If Y is a 2-subset of a 2n-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - _Milan Janjic_, Nov 18 2007
%C A051924 Prefaced with a 1: (1, 1, 4, 14, 50, ...) and convolved with the Catalan sequence = A097613: (1, 2, 7, 25, 91, ...). - _Gary W. Adamson_, May 15 2009
%C A051924 Total number of up steps before the second return in all Dyck n-paths. - _David Scambler_, Aug 21 2012
%C A051924 Conjecture: a(n) mod n^2 = n+2 iff n is an odd prime. - _Gary Detlefs_, Feb 19 2013
%C A051924 First differences of A000984 and A030662. - _J. M. Bergot_, Jun 22 2013
%C A051924 From _R. J. Mathar_, Jun 30 2013: (Start)
%C A051924 Equivalent to the Meeussen comment and the Bergot comment: The array view of A007318 is
%C A051924   1,   1,   1,   1,   1,   1,
%C A051924   1,   2,   3,   4,   5,   6,
%C A051924   1,   3,   6,  10,  15,  21,
%C A051924   1,   4,  10,  20,  35,  56,
%C A051924   1,   5,  15,  35,  70, 126,
%C A051924   1,   6,  21,  56, 126, 252,
%C A051924 and a(n) are the hook sums Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
%C A051924 From _Gus Wiseman_, Apr 12 2019: (Start)
%C A051924 Equivalent to Wouter Meeussen's comment, a(n) is the number of integer partitions (of any positive integer) such that the maximum of the length and the largest part is n. For example, the a(1) = 1 through a(3) = 14 partitions are:
%C A051924   (1)  (2)   (3)
%C A051924        (11)  (31)
%C A051924        (21)  (32)
%C A051924        (22)  (33)
%C A051924              (111)
%C A051924              (211)
%C A051924              (221)
%C A051924              (222)
%C A051924              (311)
%C A051924              (321)
%C A051924              (322)
%C A051924              (331)
%C A051924              (332)
%C A051924              (333)
%C A051924 (End)
%C A051924 Coxeter-Catalan numbers for Coxeter groups of type D_n [Armstrong]. - _N. J. A. Sloane_, Mar 09 2022
%C A051924 a(n+1) is the number of ways that a best of n pairs contest with early termination can go. For example, the first stage of an association football (soccer) penalty-kick shoot out has n=5 pairs of shots and there are a(6)=672 distinct ways it can go. For n=2 pairs, writing G for goal and M for miss, and listing the up-to-four shots in chronological order with teams alternating shots, the n(3)=14 possibilities are MMMM, MMMG, MMGM, MMGG, MGM, MGGM, MGGG, GMMM, GMMG, GMG, GGMM, GGMG, GGGM, and GGGG. Not all four shots are taken in two cases because it becomes impossible for one team to overcome the lead of the other team. - _Lee A. Newberg_, Jul 20 2024
%D A051924 Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274 16; See Table 2.8.
%H A051924 Reinhard Zumkeller, <a href="/A051924/b051924.txt">Table of n, a(n) for n = 1..1000</a>
%H A051924 Drew Armstrong, <a href="https://arxiv.org/abs/math/0611106">Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups</a>, arXiv:math/0611106 [math.CO], 2006-2007.
%H A051924 Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, <a href="https://arxiv.org/abs/2404.05672">Enumerating runs, valleys, and peaks in Catalan words</a>, arXiv:2404.05672 [math.CO], 2024. See p. 10.
%H A051924 F. Chapoton, <a href="http://irma.math.unistra.fr/~chapoton/clusters.html">Clusters</a>.
%H A051924 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000784">St000784: The maximum of the length and the largest part of the integer partition</a>.
%H A051924 Sergey Fomin and Andrei Zelevinsky, <a href="http://www.jstor.org/stable/3597238">Y-systems and generalized associahedra</a>, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
%H A051924 Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H A051924 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A051924 Joshua Marsh and Nathan Williams, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Williams/williams9.html">Nesting Nonpartitions</a>, J. Int. Seq., Vol. 25 (2022), Article 22.8.8.
%H A051924 Sanjay Moudgalya, Abhinav Prem, Rahul Nandkishore, Nicolas Regnault, and B. Andrei Bernevig, <a href="https://arxiv.org/abs/1910.14048">Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian</a>, arXiv:1910.14048 [cond-mat.str-el], 2019.
%H A051924 Hugh Thomas, <a href="http://arxiv.org/abs/math/0311334">Tamari Lattices and Non-Crossing Partitions in Types B and D</a>, arXiv:math/0311334 [math.CO], 2003-2005.
%H A051924 Lin Yang and Shengliang Yang, <a href="https://doi.org/10.4208/jms.v56n1.23.01">Protected Branches in Ordered Trees</a>, J. Math. Study, Vol. 56, No. 1 (2023), 1-17.
%F A051924 G.f.: (1-x) / sqrt(1-4*x) - 1. - _Paul D. Hanna_, Nov 08 2014
%F A051924 G.f.: Sum_{n>=1} x^n/(1-x)^(2*n) * Sum_{k=0..n} C(n,k)^2 * x^k. - _Paul D. Hanna_, Nov 08 2014
%F A051924 a(n+1) = binomial(2*n, n) + 2*Sum_{i=0..n-1} binomial(n+i, i) (V's in Pascal's Triangle). - _Jon Perry_ Apr 13 2004
%F A051924 a(n) = n*C(n-1) - (n-1)*C(n-2), where C(n) = A000108(n) = Catalan(n). For example, a(5) = 50 = 5*C(4) - 4*C(3) - 5*14 - 3*5 = 70 - 20. Triangle A128064 as an infinite lower triangular matrix * A000108 = A051924 prefaced with a 1: (1, 1, 4, 14, 50, 182, ...). - _Gary W. Adamson_, May 15 2009
%F A051924 Sum of 3 central terms of Pascal's triangle: 2*C(2+2*n, n)+C(2+2*n, 1+n). - _Zerinvary Lajos_, Dec 20 2005
%F A051924 a(n+1) = A051597(2n,n). - _Philippe Deléham_, Nov 26 2006
%F A051924 The sequence 1,1,4,... has a(n) = C(2*n,n)-C(2*(n-1),n-1) = 0^n+Sum_{k=0..n} C(n-1,k-1)*A002426(k), and g.f. given by (1-x)/(1-2*x-2*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-.... (continued fraction). - _Paul Barry_, Oct 17 2009
%F A051924 a(n) = (3*n-2)*(2*n-2)!/(n*(n-1)!^2) = A001700(n) + A001791(n-1). - _David Scambler_, Aug 21 2012
%F A051924 D-finite with recurrence: a(n) = 2*(3*n-2)*(2*n-3)*a(n-1)/(n*(3*n-5)). - _Alois P. Heinz_, Apr 25 2014
%F A051924 a(n) = 2^(-2+2*n)*Gamma(-1/2+n)*(3*n-2)/(sqrt(Pi)*Gamma(1+n)). - _Peter Luschny_, Dec 14 2015
%F A051924 a(n) ~ (3/4)*4^n*(1-(7/24)/n-(7/128)/n^2-(85/3072)/n^3-(581/32768)/n^4-(2611/262144)/n^5)/sqrt(n*Pi). - _Peter Luschny_, Dec 16 2015
%F A051924 E.g.f.: ((1 - x)*BesselI(0,2*x) + x*BesselI(1,2*x))*exp(2*x) - 1. - _Ilya Gutkovskiy_, Dec 20 2016
%F A051924 a(n) = 2 * A097613(n) for n > 1. - _Bruce J. Nicholson_, Jan 06 2019
%F A051924 Sum_{n>=1} a(n)/8^n = 7/(4*sqrt(2)) - 1. - _Amiram Eldar_, May 06 2023
%e A051924 Sums of {1}, {2, 1, 1}, {2, 2, 3, 3, 2, 1, 1}, {2, 2, 4, 5, 7, 6, 7, 5, 5, 3, 2, 1, 1}, ...
%p A051924 C:= n-> binomial(2*n, n)/(n+1): seq((n+1)*C(n)-n*C(n-1), n=1..25); # _Emeric Deutsch_, Jan 08 2008
%p A051924 Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..24); # _Zerinvary Lajos_, Jan 01 2007
%p A051924 a := n -> 2^(-2+2*n)*GAMMA(-1/2+n)*(3*n-2)/(sqrt(Pi)*GAMMA(1+n)):
%p A051924 seq(simplify(a(n)), n=1..24); # _Peter Luschny_, Dec 14 2015
%t A051924 Table[Binomial[2n,n]-Binomial[2n-2,n-1],{n,30}] (* _Harvey P. Dale_, Jan 15 2012 *)
%o A051924 (Haskell)
%o A051924 a051924 n = a051924_list !! (n-1)
%o A051924 a051924_list = zipWith (-) (tail a000984_list) a000984_list
%o A051924 -- _Reinhard Zumkeller_, May 25 2013
%o A051924 (PARI) a(n)=binomial(2*n,n)-binomial(2*n-2,n-1) \\ _Charles R Greathouse IV_, Jun 25 2013
%o A051924 (PARI) {a(n)=polcoeff((1-x) / sqrt(1-4*x +x*O(x^n)) - 1,n)}
%o A051924 for(n=1,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 08 2014
%o A051924 (PARI) {a(n)=polcoeff( sum(m=1, n, x^m * sum(k=0, m, binomial(m, k)^2 * x^k) / (1-x +x*O(x^n))^(2*m)), n)}
%o A051924 for(n=1, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Nov 08 2014
%o A051924 (Sage)
%o A051924 a = lambda n: 2^(-2+2*n)*gamma(n-1/2)*(3*n-2)/(sqrt(pi)*gamma(1+n))
%o A051924 [a(n) for n in (1..120)] # _Peter Luschny_, Dec 14 2015
%o A051924 (Magma) [Binomial(2*n, n)-Binomial(2*n-2, n-1): n in [1..28]]; // _Vincenzo Librandi_, Dec 21 2016
%Y A051924 Left-central elements of the (1, 2)-Pascal triangle A029635.
%Y A051924 Column sums of A096771.
%Y A051924 Cf. A000108, A024482 (diagonal from 2), A076540 (diagonal from 3), A000124 (row from 2), A004006 (row from 3), A006522 (row from 4).
%Y A051924 Cf. A128064; first differences of A000984.
%Y A051924 Cf. A097613.
%Y A051924 Cf. A115720, A252464, A257990, A263297, A325189, A325192, A325193.
%K A051924 easy,nice,nonn
%O A051924 1,2
%A A051924 _Barry E. Williams_, Dec 19 1999
%E A051924 Edited by _N. J. A. Sloane_, May 03 2008, at the suggestion of _R. J. Mathar_