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A000588 a(n) = 7*binomial(2n,n-3)/(n+4).
(Formerly M4413 N1866)

%I M4413 N1866

%S 0,0,0,1,7,35,154,637,2548,9996,38760,149226,572033,2187185,8351070,

%T 31865925,121580760,463991880,1771605360,6768687870,25880277150,

%U 99035193894,379300783092,1453986335186,5578559816632,21422369201800,82336410323440,316729578421620

%N a(n) = 7*binomial(2n,n-3)/(n+4).

%C a(n-5) is the number of n-th generation vertices in the tree of sequences with unit increase labeled by 6 (cf. _Zoran Sunic_ reference). - _Benoit Cloitre_, Oct 07 2003

%C 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=3. Example: For n=3 there is only one path EEENNN. - _Herbert Kociemba_, May 24 2004

%C Number of standard tableaux of shape (n+3,n-3). - _Emeric Deutsch_, May 30 2004

%C a(n) = A214292(2*n-1,n-4) for n > 3. - _Reinhard Zumkeller_, Jul 12 2012

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A000588/b000588.txt">Table of n, a(n) for n = 0..200</a>

%H D. E. Davenport, L. K. Pudwell, L. W. Shapiro, L. C. Woodson, <a href="http://faculty.valpo.edu/lpudwell/papers/treeboundary.pdf">The Boundary of Ordered Trees</a>, 2014.

%H H. H. Gudmundsson, <a href="http://puma.dimai.unifi.it/21_2/9_Gudmundsson.pdf">Dyck paths, standard Young tableaux, and pattern avoiding permutations</a>, PU. M. A. Vol. 21 (2010), No. 2, pp. 265-284 (see 4.4 p. 279).

%H R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

%H V. E. Hoggatt, Jr. and M. Bicknell, <a href="http://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.

%H A. Papoulis, <a href="/A000108/a000108_8.pdf">A new method of inversion of the Laplace transform</a>, Quart. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]

%H A. Papoulis, <a href="http://www.jstor.org/stable/43636019">A new method of inversion of the Laplace transform</a>, Quart. Applied Math. 14 (1956), 405ff.

%H J. Riordan, <a href="/A000262/a000262_1.pdf">Letter to N. J. A. Sloane, Nov 10 1970</a>

%H J. Riordan, <a href="http://www.jstor.org/stable/2005477">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., 29 (1975), 215-222.

%H _Zoran Sunic_, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1n5">Self-Describing Sequences and the Catalan Family Tree</a>, Electronic Journal of Combinatorics, 10 (2003) #N5.

%F Expansion of x^3*C^7, where C = (1-(1-4*x)^(1/2))/(2*x) is the g.f. for the Catalan numbers, A000108. - _Philippe Deléham_, Feb 03 2004

%F 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>=6, a(n-3)=(-1)^(n-6)*coeff(charpoly(A,x),x^6). [_Milan Janjic_, Jul 08 2010]

%F (n+4)*a(n) +(-9*n-20)*a(n-1) +2*(13*n+5)*a(n-2) +(-25*n+38)*a(n-3) +2*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Jun 20 2013

%F From _Ilya Gutkovskiy_, Jan 22 2017: (Start)

%F E.g.f.: (1/6)*x^3*1F1(7/2; 8; 4*x).

%F a(n) ~ 7*4^n/(sqrt(Pi)*n^(3/2)). (End)

%F 0 = a(n)*(+1456*a(n+1) - 87310*a(n+2) + 132834*a(n+3) - 68068*a(n+4) + 9724*a(n+5)) + a(n+1)*(+8918*a(n+1) - 39623*a(n+2) + 51726*a(n+3) - 299*a(n+4) - 1573*a(n+5)) + a(n+2)*(-24696*a(n+2) - 1512*a(n+3) + 1008*a(n+4)) for all n in Z. - _Michael Somos_, Jan 22 2017

%e G.f. = x^3 + 7*x^4 + 35*x^5 + 154*x^6 + 637*x^7 + 2548*x^8 + 9996*x^9 + ...

%o (PARI) A000588(n)=7*binomial(2*n,n-3)/(n+4) \\ _M. F. Hasler_, Aug 25 2012

%o (PARI) x='x+O('x^50); concat([0, 0, 0], Vec(x^3*((1-(1-4*x)^(1/2))/(2*x))^7)) \\ _Altug Alkan_, Nov 01 2015

%Y First differences are in A026014.

%Y A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

%Y Cf. A000108, A000245, A002057, A000344, A003517, A003518, A003519, A001392.

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_

%E More terms from _N. J. A. Sloane_, Jul 13 2010

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Last modified January 20 12:50 EST 2019. Contains 319330 sequences. (Running on oeis4.)