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

%I A000588 M4413 N1866 #121 Jan 05 2025 19:51:31
%S A000588 0,0,0,1,7,35,154,637,2548,9996,38760,149226,572033,2187185,8351070,
%T A000588 31865925,121580760,463991880,1771605360,6768687870,25880277150,
%U A000588 99035193894,379300783092,1453986335186,5578559816632,21422369201800,82336410323440,316729578421620
%N A000588 a(n) = 7*binomial(2n,n-3)/(n+4).
%C A000588 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 A000588 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 A000588 Number of standard tableaux of shape (n+3,n-3). - _Emeric Deutsch_, May 30 2004
%D A000588 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000588 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000588 T. D. Noe, <a href="/A000588/b000588.txt">Table of n, a(n) for n = 0..200</a>
%H A000588 Joerg Arndt, <a href="/A000588/a000588.txt">The a(5)=35 Young tableaux of shape [8,2]</a>.
%H A000588 Dennis E. Davenport, Louis W. Shapiro, Lara K. Pudwell and Leon C. Woodson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Davenport/dav3.html">The Boundary of Ordered Trees</a>, J. Integer Seq., Vol. 18 (2015), Article 15.5.8; <a href="http://faculty.valpo.edu/lpudwell/papers/treeboundary.pdf">alternative link</a>.
%H A000588 Hilmar Haukur 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, No. 2 (2010), pp. 265-284 (see 4.4 p. 279).
%H A000588 Richard 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 A000588 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
%H A000588 Athanasios Papoulis, <a href="/A000108/a000108_8.pdf">A new method of inversion of the Laplace transform</a>, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]
%H A000588 Athanasios Papoulis, <a href="http://www.jstor.org/stable/43636019">A new method of inversion of the Laplace transform</a>, Quart. Applied Math., Vol. 14 (1956), pp. 405-414.
%H A000588 John Riordan, <a href="/A000262/a000262_1.pdf">Letter to N. J. A. Sloane, Nov 10 1970</a>.
%H A000588 John Riordan, <a href="https://doi.org/10.1090/S0025-5718-1975-0366686-9">The distribution of crossings of chords joining pairs of 2n points on a circle</a>, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.
%H A000588 Zoran Sunic, <a href="https://doi.org/10.37236/1745">Self-Describing Sequences and the Catalan Family Tree</a>, Electronic Journal of Combinatorics, Vol. 10 (2003), Article N5.
%F A000588 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 A000588 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 A000588 a(n) = A214292(2*n-1,n-4) for n > 3. - _Reinhard Zumkeller_, Jul 12 2012
%F A000588 From _Ilya Gutkovskiy_, Jan 22 2017: (Start)
%F A000588 E.g.f.: (1/6)*x^3*1F1(7/2; 8; 4*x).
%F A000588 a(n) ~ 7*4^n/(sqrt(Pi)*n^(3/2)). (End)
%F A000588 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
%F A000588 From _Amiram Eldar_, Jan 02 2022: (Start)
%F A000588 Sum_{n>=3} 1/a(n) = 27/14 - 26*Pi/(63*sqrt(3)).
%F A000588 Sum_{n>=3} (-1)^(n+1)/a(n) = 11364*log(phi)/(175*sqrt(5)) - 4583/350, where phi is the golden ratio (A001622). (End)
%F A000588 a(n) = Integral_{x=0..4} x^(n)*W(x)dx, n>=0, where W(x) = sqrt(4/x - 1)*(x^3 - 5*x^2 + 6*x - 1)/(2*Pi). The function W(x) for x->0 tends to -infinity (which is its absolute minimum), and W(4) = 0. W(x) is a signed function on the interval x = (0, 4) where it has two maxima separated by one local minimum. - _Karol A. Penson_, Jun 17 2024
%F A000588 D-finite with recurrence -(n+4)*(n-3)*a(n) +2*n*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Jul 30 2024
%F A000588 a(n) = A000108(n+3) - 5*A000108(n+2) + 6*A000108(n+1) - A000108(n). - _Taras Goy_, Dec 21 2024
%e A000588 G.f. = x^3 + 7*x^4 + 35*x^5 + 154*x^6 + 637*x^7 + 2548*x^8 + 9996*x^9 + ...
%t A000588 a[n_] := 7*Binomial[2n, n-3]/(n + 4); Table[a[n],{n,0,27}] (* _James C. McMahon_, Dec 05 2023 *)
%o A000588 (PARI) A000588(n)=7*binomial(2*n,n-3)/(n+4) \\ _M. F. Hasler_, Aug 25 2012
%o A000588 (PARI) my(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 A000588 First differences are in A026014.
%Y A000588 A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
%Y A000588 Cf. A000108, A000245, A002057, A000344, A003517, A003518, A003519, A001392, A001622.
%K A000588 nonn,easy
%O A000588 0,5
%A A000588 _N. J. A. Sloane_
%E A000588 More terms from _N. J. A. Sloane_, Jul 13 2010