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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Examples

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

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.

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.

Programs

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 *)

PARI

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

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)

Extensions

More terms from Harvey P. Dale, Mar 03 2011