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A051924
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a(n) = 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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39
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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
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OFFSET
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1,2
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COMMENTS
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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
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 ... (End)
This sequence also gives the number of clusters and non-crossing partitions of type D_n. - F. Chapoton, 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 Janjic, Nov 18 2007
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
Total number of up steps before the second return in all Dyck n-paths. - David Scambler, Aug 21 2012
Conjecture: a(n) mod n^2 = n+2 iff n is an odd prime. - Gary Detlefs, Feb 19 2013
Equivalent to the Meeussen comment and the Bergot comment: The array view of A007318 is
1, 1, 1, 1, 1, 1,
1, 2, 3, 4, 5, 6,
1, 3, 6, 10, 15, 21,
1, 4, 10, 20, 35, 56,
1, 5, 15, 35, 70, 126,
1, 6, 21, 56, 126, 252,
and a(n) are the hook sums Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
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 k. For example, the a(1) = 1 through a(3) = 14 partitions are:
(1) (2) (3)
(11) (31)
(21) (32)
(22) (33)
(111)
(211)
(221)
(222)
(311)
(321)
(322)
(331)
(332)
(333)
(End)
Coxeter-Catalan numbers for Coxeter groups of type D_n [Armstrong]. - N. J. A. Sloane, Mar 09 2022
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REFERENCES
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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. (Also https://arxiv.org/pdf/math/0611106.pdf)
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LINKS
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FORMULA
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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
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
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
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
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
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
a(n) = 2^(-2+2*n)*Gamma(-1/2+n)*(3*n-2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
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
E.g.f.: ((1 - x)*BesselI(0,2*x) + x*BesselI(1,2*x))*exp(2*x) - 1. - Ilya Gutkovskiy, Dec 20 2016
Sum_{n>=1} a(n)/8^n = 7/(4*sqrt(2)) - 1. - Amiram Eldar, May 06 2023
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EXAMPLE
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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
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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
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
a := n -> 2^(-2+2*n)*GAMMA(-1/2+n)*(3*n-2)/(sqrt(Pi)*GAMMA(1+n)):
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MATHEMATICA
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Table[Binomial[2n, n]-Binomial[2n-2, n-1], {n, 30}] (* Harvey P. Dale, Jan 15 2012 *)
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PROG
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(Haskell)
a051924 n = a051924_list !! (n-1)
a051924_list = zipWith (-) (tail a000984_list) a000984_list
(PARI) {a(n)=polcoeff((1-x) / sqrt(1-4*x +x*O(x^n)) - 1, n)}
(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)}
(Sage)
a = lambda n: 2^(-2+2*n)*gamma(n-1/2)*(3*n-2)/(sqrt(pi)*gamma(1+n))
(Magma) [Binomial(2*n, n)-Binomial(2*n-2, n-1): n in [1..28]]; // Vincenzo Librandi, Dec 21 2016
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CROSSREFS
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Left-central elements of the (1, 2)-Pascal triangle A029635.
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KEYWORD
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easy,nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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