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A000344 a(n) = 5*binomial(2n,n-2)/(n+3).
(Formerly M3904 N1602)
32
1, 5, 20, 75, 275, 1001, 3640, 13260, 48450, 177650, 653752, 2414425, 8947575, 33266625, 124062000, 463991880, 1739969550, 6541168950, 24647883000, 93078189750, 352207870014, 1335293573130, 5071418015120, 19293438101000, 73514652074500, 280531912316292 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

a(n-3) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 4 (cf. Zoran Sunik 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=2. Example: For n=3 there are the 5 paths EENENN, EENNEN, EENNNE, ENEENN, NEEENN. - Herbert Kociemba, May 24 2004

Number of standard tableaux of shape (n+2,n-2). - Emeric Deutsch, May 30 2004

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

REFERENCES

C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146.

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

Vincenzo Librandi, Table of n, a(n) for n = 2..170

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Dennis E. Davenport, Lara K. Pudwell, Louis W. Shapiro, Leon C. Woodson, The Boundary of Ordered Trees, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.8.

H. H. Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A. Vol. 21 (2010), No. 2, pp. 265-284 (see Theorem 4.2 p. 275).

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.

C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146. [Annotated scanned copy]

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, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.

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

FORMULA

Integral representation as n-th moment of a function on [0, 4], in Maple notation: a(n)=int(x^n*((1/2)/Pi*x^(3/2)*(x^2-3*x+1)*(4-x)^(1/2)), x=0..4), n=0, 1..., for which offset=0. - Karol A. Penson, Oct 11 2001

Expansion of x^2*C^5, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers (A000108). - Herbert Kociemba, May 02 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>=4, a(n-2)=(-1)^(n-4)*coeff(charpoly(A,x),x^4). - Milan Janjic, Jul 08 2010

a(n) = A000108(n+2) - 3*A000108(n+1)+ A000108(n). - David Scambler, May 20 2012

(n+3)*(n-2)*a(n) = 2*n*(2n-1)*a(n-1). - R. J. Mathar, Jun 27 2012

0 = a(n)*(-528*a(n+1) + 9162*a(n+2) - 9295*a(n+3) + 1859*a(n+4)) + a(n+1)*(-1650*a(n+1) - 762*a(n+2) + 4188*a(n+3) - 946*a(n+4)) + a(n+2)*(-1050*a(n+2) - 126*a(n+3) + 84*a(n+4)) for all n in Z. - Michael Somos, May 28 2014

0 = a(n)*(a(n)*(+16*a(n+1) + 6*a(n+2)) + a(n+1)*(+66*a(n+1) - 105*a(n+2) + 40*a(n+3)) + a(n+2)*(-69*a(n+2) + 15*a(n+3))) +a(n+1)*(a(n+1)*(50*a(n+1) + 42*a(n+2) - 28*a(n+3)) +a(n+2)*(+12*a(n+2))) for all n in Z. - Michael Somos, May 28 2014

0 = a(n)^2*(-16*a(n+1)^2 - 38*a(n+1)*a(n+2) - 12*a(n+2)^2) + a(n)*a(n+1)*(-66*a(n+1)^2 + 149*a(n+1)*a(n+2) - 23*a(n+2)^2) + a(n+1)^2*(-50*a(n+1)^2 + 2*a(n+2)^2) for all n in Z. - Michael Somos, May 28 2014

From Ilya Gutkovskiy, Jan 22 2017: (Start)

E.g.f.: (x*(2 + x) * BesselI(0, 2*x) - (2+x+x^2) * BesselI(1, 2*x)) * exp(2*x)/x^2.

a(n) ~ 5*4^n/(sqrt(Pi)*n^(3/2)). (End)

EXAMPLE

G.f. = x^2 + 5*x^3 + 20*x^4 + 75*x^5 + 275*x^6 + 1001*x^7 + 3640*x^8 + ...

MATHEMATICA

Table[5 Binomial[2n, n-2]/(n+3), {n, 2, 40}] (* or *) CoefficientList[Series[ (1-Sqrt[1-4 x]+x (-5+3 Sqrt[1-4 x]-(-5+Sqrt[1-4 x]) x))/(2 x^5), {x, 0, 38}], x]  (* Harvey P. Dale, May 01 2011 *)

a[ n_] := If[ n < 0, 0, 5 Binomial[2 n, n - 2] / (n + 3)]; (* Michael Somos, May 28 2014 *)

PROG

(MAGMA) [5*Binomial(2*n, n-2)/(n+3): n in [2..30]]; // Vincenzo Librandi, May 03 2011

(PARI) a(n)=5*binomial(2*n, n-2)/(n+3) \\ Charles R Greathouse IV, Jul 25 2011

CROSSREFS

T(n, n+5) for n=0, 1, 2, ..., array T as in A047072.

A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Cf. A000108, A000245, A002057, A003517, A000588, A003518, A003519, A001392.

Sequence in context: A093131 A030191 A224422 * A275909 A275908 A270023

Adjacent sequences:  A000341 A000342 A000343 * A000345 A000346 A000347

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 28 15:38 EDT 2017. Contains 284243 sequences.