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A000588
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a(n) = 7*binomial(2n,n-3)/(n+4).
(Formerly M4413 N1866)
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27
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0, 0, 0, 1, 7, 35, 154, 637, 2548, 9996, 38760, 149226, 572033, 2187185, 8351070, 31865925, 121580760, 463991880, 1771605360, 6768687870, 25880277150, 99035193894, 379300783092, 1453986335186, 5578559816632, 21422369201800, 82336410323440, 316729578421620
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OFFSET
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0,5
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COMMENTS
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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
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
Number of standard tableaux of shape (n+3,n-3). - Emeric Deutsch, May 30 2004
a(n) = A214292(2*n-1,n-4) for n > 3. - Reinhard Zumkeller, Jul 12 2012
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
D. E. Davenport, L. K. Pudwell, L. W. Shapiro, L. C. Woodson, The Boundary of Ordered Trees, 2014.
H. H. Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A. Vol. 21 (2010), No. 2, pp. 265-284 (see 4.4 p. 279).
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
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. Appl. Math 14 (1957), 405-414. [Annotated scan of selected pages]
A. Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math. 14 (1956), 405ff.
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1970
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
_Zoran Sunic_, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, 10 (2003) #N5.
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FORMULA
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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
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]
D-finite with recurrence: (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
From Ilya Gutkovskiy, Jan 22 2017: (Start)
E.g.f.: (1/6)*x^3*1F1(7/2; 8; 4*x).
a(n) ~ 7*4^n/(sqrt(Pi)*n^(3/2)). (End)
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
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EXAMPLE
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G.f. = x^3 + 7*x^4 + 35*x^5 + 154*x^6 + 637*x^7 + 2548*x^8 + 9996*x^9 + ...
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PROG
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(PARI) A000588(n)=7*binomial(2*n, n-3)/(n+4) \\ M. F. Hasler, Aug 25 2012
(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
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CROSSREFS
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First differences are in A026014.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A002057, A000344, A003517, A003518, A003519, A001392.
Sequence in context: A022635 A336602 A217274 * A005285 A006095 A171477
Adjacent sequences: A000585 A000586 A000587 * A000589 A000590 A000591
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from N. J. A. Sloane, Jul 13 2010
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STATUS
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approved
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