A001392 a(n) = 9*binomial(2n,n-4)/(n+5). (Formerly M4637 N1981)

1, 9, 54, 273, 1260, 5508, 23256, 95931, 389367, 1562275, 6216210, 24582285, 96768360, 379629720, 1485507600, 5801732460, 22626756594, 88152205554, 343176898988, 1335293573130, 5193831553416, 20198233818840, 78542105700240, 305417807763705

Offset

4,2

Comments

Number of n-th generation vertices in the tree of sequences with unit increase labeled by 8 (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=4. - Herbert Kociemba, May 24 2004

Number of standard tableaux of shape (n+4,n-4). - 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 = 4..200

Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

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 (1957), pp. 405-414.

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^4*C^9, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for 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>=8, a(n-4)=(-1)^(n-8)*coeff(charpoly(A,x),x^8). - Milan Janjic, Jul 08 2010

a(n) = A214292(2*n-1,n-5) for n > 4. - Reinhard Zumkeller, Jul 12 2012

D-finite with recurrence -(n+5)*(n-4)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jun 20 2013

From Ilya Gutkovskiy, Jan 22 2017: (Start)

E.g.f.: (1/24)*x^4*1F1(9/2; 10; 4*x).

a(n) ~ 9*4^n/(sqrt(Pi)*n^(3/2)). (End)

From Amiram Eldar, Jan 02 2022: (Start)

Sum_{n>=4} 1/a(n) = 158*Pi/(81*sqrt(3)) - 649/270.

Sum_{n>=4} (-1)^n/a(n) = 52076*log(phi)/(225*sqrt(5)) - 22007/450, where phi is the golden ratio (A001622). (End)

Example

G.f. = x^4 + 9*x^5 + 54*x^6 + 273*x^7 + 1260*x^8 + 5508*x^9 + 23256*x^10 + ...

Maple

A001392:=n->9*binomial(2*n, n-4)/(n+5): seq(A001392(n), n=4..40); # Wesley Ivan Hurt, Apr 11 2017

Mathematica

Table[9*Binomial[2n, n-4]/(n+5), {n, 4, 30}] (* Harvey P. Dale, Mar 03 2011 *)

Prog

(PARI) a(n)=9*binomial(n+n, n-4)/(n+5) \\ Charles R Greathouse IV, Jul 31 2011

Crossrefs

First differences are in A026015.

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, A001622.

Sequence in context: A359722 A027472 A022637 * A188428 A243415 A276602

Adjacent sequences: A001389 A001390 A001391 * A001393 A001394 A001395

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Harvey P. Dale, Mar 03 2011

Status

approved