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A000344
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5*binomial(2n,n-2)/(n+3).
(Formerly M3904 N1602)
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28
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1, 5, 20, 75, 275, 1001, 3640, 13260, 48450, 177650, 653752, 2414425, 8947575, 33266625, 124062000, 463991880, 1739969550, 6541168950, 24647883000, 93078189750, 352207870014, 1335293573130, 5071418015120, 19293438101000, 73514652074500, 280531912316292
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| a(n-3) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 4 (cf. Zoran Sunik reference) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=2 Example: For n=3 there are the 5 paths EENENN, EENNEN, EENNNE, ENEENN, NEEENN - Herbert Kociemba (kociemba(AT)t-online.de), May 24 2004
Number of standard tableaux of shape (n+2,n-2). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 30 2004
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REFERENCES
| Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math. 14 (1956), 405ff.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 2..170
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
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FORMULA
| Integral representation as n-th moment of a function on [0, 4], in Maple notation: a(n)=int(x^n*((1/2)/Pi*x^(3/2)*(x^2-3*x+1)*(4-x)^(1/2)), x=0..4), n=0, 1..., for which offset=0. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Oct 11 2001
Expansion of x^2*C^5, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers (A000108) - Herbert Kociemba (kociemba(AT)t-online.de), May 02 2004
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=4, a(n-2)=(-1)^(n-4)*coeff(charpoly(A,x),x^4). [From Milan R. Janjic (agnus(AT)blic.net), Jul 08 2010]
Conjecture: n*(n+5)*a(n) +(n^2+n-32)*a(n-1) -10*(n+1)*(2n+1)*a(n-2)=0. - R. J. Mathar, Dec 17 2011
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MATHEMATICA
| Table[5 Binomial[2n, n-2]/(n+3), {n, 2, 40}] (* or *) CoefficientList[Series[ (1-Sqrt[1-4 x]+x (-5+3 Sqrt[1-4 x]-(-5+Sqrt[1-4 x]) x))/(2 x^5), {x, 0, 38}], x] (* From Harvey P. Dale, May 01 2011 *)
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PROG
| (MAGMA) [5*Binomial(2*n, n-2)/(n+3): n in [2..30]]; // Vincenzo Librandi, May 03 2011
(PARI) a(n)=5*binomial(2*n, n-2)/(n+3) \\ Charles R Greathouse IV, Jul 25 2011
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CROSSREFS
| T(n, n+5) for n=0, 1, 2, ..., array T as in A047072.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392.
Sequence in context: A094828 A030191 A093131 * A061278 A000758 A005283
Adjacent sequences: A000341 A000342 A000343 * A000345 A000346 A000347
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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