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A051924
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Binomial(2*n,n)-binomial(2*n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).
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12
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1, 4, 14, 50, 182, 672, 2508, 9438, 35750, 136136, 520676, 1998724, 7696444, 29716000, 115000920, 445962870, 1732525830, 6741529080, 26270128500, 102501265020, 400411345620, 1565841089280, 6129331763880, 24014172955500
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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 (wouter.meeussen(AT)pandora.be), Dec 10 2001
From Benoit Cloitre, Jan 29 2002: 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):
1 2 3 4 .....
2 4 7 11 .......
3 7 14 25 .......
4 11 25 50 .......
This sequence also gives the number of clusters and non-crossing partitions of type D_n. - Frederic Chapoton (fchapoton(AT)voila.fr), Jan 31 2005
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 R. Janjic (agnus(AT)blic.net), Nov 18 2007
Prefaced with a 1: (1, 1, 4, 14, 50,...) and convolved with the Catalan sequence = A097613: (1, 2, 7, 25, 91,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2009]
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REFERENCES
| Fomin, Sergey and Zelevinsky, Andrei, Y-systems and generalized associahedra, Ann. of Math. (2) 158,2003.
Hugh Thomas, math.CO/0311334: Tamari Lattices and Non-Crossing Partitions in Types B and D.
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LINKS
| F. Chapoton, Clusters.
Milan Janjic, Two Enumerative Functions
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FORMULA
| 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 (perry(AT)globalnet.co.uk) Apr 13 2004
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 (qntmpkt(AT)yahoo.com), May 15 2009
Sum of 3 central terms of Pascal's triangle: 2*C(2+2*n, n)+C(2+2*n, 1+n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 20 2005
a(n+1)=A051597(2n,n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 26 2006
Contribution from Paul Barry (pbarry(AT)wit.ie), Oct 17 2009: (Start)
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). (End)
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EXAMPLE
| 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}, ...
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MAPLE
| 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 (deutsch(AT)duke.poly.edu), Jan 08 2008
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 (zerinvarylajos(AT)yahoo.com), Jan 01 2007
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MATHEMATICA
| Table[Binomial[2n, n]-Binomial[2n-2, n-1], {n, 30}] (* From Harvey P. Dale, Jan 15 2012 *)
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CROSSREFS
| Left-central elements of the (1, 2)-Pascal triangle A029635.
Cf. A000108, A024482 (diagonal from 2), A076540 (diagonal from 3), A000124 (row from 2), A004006 (row from 3), A006522 (row from 4).
Cf. A128064.
Sequence in context: A047065 A055990 A087945 * A076024 A062807 A117421
Adjacent sequences: A051921 A051922 A051923 * A051925 A051926 A051927
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KEYWORD
| easy,nice,nonn
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AUTHOR
| Barry E. Williams, Dec 19 1999
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), May 03 2008, at the suggestion of R. J. Mathar.
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