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

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

Offset

0,5

Comments

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

References

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).

Links

T. D. Noe, Table of n, a(n) for n = 0..200

Joerg Arndt, The a(5)=35 Young tableaux of shape [8,2].

Dennis E. Davenport, Louis W. Shapiro, Lara K. Pudwell and Leon C. Woodson, The Boundary of Ordered Trees, J. Integer Seq., Vol. 18 (2015), Article 15.5.8; alternative link.

Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21, No. 2 (2010), pp. 265-284 (see 4.4 p. 279).

Richard 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., Vol. 14, No. 5 (1976), pp. 395-405.

Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math., Vol. 14 (1957), pp. 405-414. [Annotated scan of selected pages]

Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Applied Math., Vol. 14 (1956), pp. 405-414.

John Riordan, Letter to N. J. A. Sloane, Nov 10 1970.

John Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., Vol. 29, No. 129 (1975), pp. 215-222.

Zoran Sunic, Self-Describing Sequences and the Catalan Family Tree, Electronic Journal of Combinatorics, Vol. 10 (2003), Article N5.

Formula

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

a(n) = A214292(2*n-1,n-4) for n > 3. - Reinhard Zumkeller, Jul 12 2012

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

From Amiram Eldar, Jan 02 2022: (Start)

Sum_{n>=3} 1/a(n) = 27/14 - 26*Pi/(63*sqrt(3)).

Sum_{n>=3} (-1)^(n+1)/a(n) = 11364*log(phi)/(175*sqrt(5)) - 4583/350, where phi is the golden ratio (A001622). (End)

Example

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

Mathematica

a[n_] := 7*Binomial[2n, n-3]/(n + 4); Table[a[n], {n, 0, 27}] (* James C. McMahon, Dec 05 2023 *)

Prog

(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

Crossrefs

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, A001622.

Sequence in context: A336602 A370391 A217274 * A005285 A371964 A006095

Adjacent sequences: A000585 A000586 A000587 * A000589 A000590 A000591

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from N. J. A. Sloane, Jul 13 2010

Status

approved