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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). 21

%I

%S 1,4,14,50,182,672,2508,9438,35750,136136,520676,1998724,7696444,

%T 29716000,115000920,445962870,1732525830,6741529080,26270128500,

%U 102501265020,400411345620,1565841089280,6129331763880,24014172955500

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

%C 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 From _Benoit Cloitre_, Jan 29 2002: (Start)

%C 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 1 2 3 4 ...

%C 2 4 7 11 ...

%C 3 7 14 25 ...

%C 4 11 25 50 ... (End)

%C This sequence also gives the number of clusters and non-crossing partitions of type D_n. - _F. Chapoton_, Jan 31 2005

%C 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 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 Total number of up steps before the second return in all Dyck n-paths. - _David Scambler_, Aug 21 2012

%C Conjecture: a(n) mod n^2 = n+2 iff n is an odd prime. - _Gary Detlefs_, Feb 19 2013

%C First differences of A000984 and A030662. - _J. M. Bergot_, Jun 22 2013

%C From _R. J. Mathar_, Jun 30 2013: (Start)

%C Equivalent to the Meeussen comment and the Bergot comment: The array view of A007318 is

%C 1, 1, 1, 1, 1, 1,

%C 1, 2, 3, 4, 5, 6,

%C 1, 3, 6, 10, 15, 21,

%C 1, 4, 10, 20, 35, 56,

%C 1, 5, 15, 35, 70, 126,

%C 1, 6, 21, 56, 126, 252,

%C and a(n) are the hook sums Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)

%H Reinhard Zumkeller, <a href="/A051924/b051924.txt">Table of n, a(n) for n = 1..1000</a>

%H F. Chapoton, <a href="http://irma.math.unistra.fr/~chapoton/clusters.html">Clusters</a>.

%H 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 Joël Gay, Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>

%H 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.

%F G.f.: (1-x) / sqrt(1-4*x) - 1. - _Paul D. Hanna_, Nov 08 2014

%F 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 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 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 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 a(n+1) = A051597(2n,n). - _Philippe Deléham_, Nov 26 2006

%F The sequence 1,1,4,... has a(n)=C(2n,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)/(1-2x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-.... (continued fraction). - _Paul Barry_, Oct 17 2009

%F a(n) = (3*n-2)*(2*n-2)!/(n*(n-1)!^2) = A001700(n) + A001791(n-1). - _David Scambler_, Aug 21 2012

%F a(n) = 2*(3*n-2)*(2*n-3)*a(n-1)/(n*(3*n-5)). - _Alois P. Heinz_, Apr 25 2014

%F a(n) = 2^(-2+2*n)*Gamma(-1/2+n)*(3*n-2)/(sqrt(Pi)*Gamma(1+n)). - _Peter Luschny_, Dec 14 2015

%F 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 E.g.f.: ((1 - x)*BesselI(0,2*x) + x*BesselI(1,2*x))*exp(2*x) - 1. - _Ilya Gutkovskiy_, Dec 20 2016

%e 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 C:=proc(n) options operator, arrow: binomial(2*n, n)/(n+1) end proc: seq(n*C(n-1)-(n-1)*C(n-2),n=2..25); # _Emeric Deutsch_, Jan 08 2008

%p 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 a := n -> 2^(-2+2*n)*GAMMA(-1/2+n)*(3*n-2)/(sqrt(Pi)*GAMMA(1+n)):

%p seq(simplify(a(n)), n=1..24); # _Peter Luschny_, Dec 14 2015

%t Table[Binomial[2n,n]-Binomial[2n-2,n-1],{n,30}] (* _Harvey P. Dale_, Jan 15 2012 *)

%o (Haskell)

%o a051924 n = a051924_list !! (n-1)

%o a051924_list = zipWith (-) (tail a000984_list) a000984_list

%o -- _Reinhard Zumkeller_, May 25 2013

%o (PARI) a(n)=binomial(2*n,n)-binomial(2*n-2,n-1) \\ _Charles R Greathouse IV_, Jun 25 2013

%o (PARI) {a(n)=polcoeff((1-x) / sqrt(1-4*x +x*O(x^n)) - 1,n)}

%o for(n=1,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 08 2014

%o (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 for(n=1, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Nov 08 2014

%o (Sage)

%o a = lambda n: 2^(-2+2*n)*gamma(n-1/2)*(3*n-2)/(sqrt(pi)*gamma(1+n))

%o [a(n) for n in (1..120)] # _Peter Luschny_, Dec 14 2015

%o (MAGMA) [Binomial(2*n, n)-Binomial(2*n-2, n-1): n in [1..28]]; // _Vincenzo Librandi_, Dec 21 2016

%Y Left-central elements of the (1, 2)-Pascal triangle A029635.

%Y Cf. A000108, A024482 (diagonal from 2), A076540 (diagonal from 3), A000124 (row from 2), A004006 (row from 3), A006522 (row from 4).

%Y Cf. A128064; first differences of A000984.

%K easy,nice,nonn

%O 1,2

%A _Barry E. Williams_, Dec 19 1999

%E Edited by _N. J. A. Sloane_, May 03 2008, at the suggestion of _R. J. Mathar_

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Last modified September 23 02:13 EDT 2018. Contains 315271 sequences. (Running on oeis4.)