login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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
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; text; internal format)
OFFSET

1,2

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, Dec 10 2001

From Benoit Cloitre, Jan 29 2002: (Start)

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

First differences of A000984 and A030662. - J. M. Bergot, Jun 22 2013

From R. J. Mathar, Jun 30 2013: (Start)

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)

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

F. Chapoton, Clusters.

Sergey Fomin and Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

Milan Janjic, Two Enumerative Functions

Hugh Thomas, Tamari Lattices and Non-Crossing Partitions in Types B and D, arXiv:math/0311334 [math.CO], 2003-2005.

FORMULA

G.f.: (1-x) / sqrt(1-4*x) - 1. - Paul D. Hanna, Nov 08 2014

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

a(n+1) = A051597(2n,n). - Philippe Deléham, Nov 26 2006

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

a(n) = (3*n-2)*(2*n-2)!/(n*(n-1)!^2) = A001700(n) + A001791(n-1). - David Scambler, Aug 21 2012

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

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}, ...

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, 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)):

seq(simplify(a(n)), n=1..24); # Peter Luschny, Dec 14 2015

MATHEMATICA

Table[Binomial[2n, n]-Binomial[2n-2, n-1], {n, 30}] (* Harvey P. Dale, Jan 15 2012 *)

PROG

(Haskell)

a051924 n = a051924_list !! (n-1)

a051924_list = zipWith (-) (tail a000984_list) a000984_list

-- Reinhard Zumkeller, May 25 2013

(PARI) a(n)=binomial(2*n, n)-binomial(2*n-2, n-1) \\ Charles R Greathouse IV, Jun 25 2013

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

for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 08 2014

(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)}

for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 08 2014

(Sage)

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

[a(n) for n in (1..120)] # Peter Luschny, Dec 14 2015

(MAGMA) [Binomial(2*n, n)-Binomial(2*n-2, n-1): n in [1..28]]; // Vincenzo Librandi, Dec 21 2016

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; first differences of A000984.

Sequence in context: A055990 A211308 A087945 * A272687 A076024 A062807

Adjacent sequences:  A051921 A051922 A051923 * A051925 A051926 A051927

KEYWORD

easy,nice,nonn

AUTHOR

Barry E. Williams, Dec 19 1999

EXTENSIONS

Edited by N. J. A. Sloane, May 03 2008, at the suggestion of R. J. Mathar

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 25 09:07 EDT 2018. Contains 304560 sequences. (Running on oeis4.)