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A003519 a(n) = 10*C(2n+1, n-4)/(n+6).
(Formerly M4721)
23
1, 10, 65, 350, 1700, 7752, 33915, 144210, 600875, 2466750, 10015005, 40320150, 161280600, 641886000, 2544619500, 10056336264, 39645171810, 155989499540, 612815891050, 2404551645100, 9425842448792, 36921502679600, 144539291740025, 565588532895750, 2212449261033375 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

COMMENTS

Number of standard tableaux of shape (n+5,n-4). - Emeric Deutsch, May 30 2004

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

a(n) is the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly twice. By symmetry, it is also the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x vertically exactly twice. Details can be found in Section 3.3 in Pan and Remmel's link. - Ran Pan, Feb 02 2016

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Robert Israel, Table of n, a(n) for n = 4..1650

R. K. Guy, Letter to N. J. A. Sloane, May 1990

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

V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

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.

Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

FORMULA

G.f.: x^4*C(x)^10, where C(x)=[1-sqrt(1-4x)]/(2x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 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>=9, a(n-5)=(-1)^(n-9)*coeff(charpoly(A,x),x^9). [Milan Janjic, Jul 08 2010]

From Robert Israel, Feb 02 2016: (Start)

a(n+1) = 2*(n+1)*(2n+3)/((n+7)*(n-3)) * a(n).

a(n) ~ 20 * 4^n/sqrt(Pi*n^3). (End)

E.g.f.: 5*BesselI(5,2*x)*exp(2*x)/x. - Ilya Gutkovskiy, Jan 23 2017

MAPLE

seq(10*binomial(2*n+1, n-4)/(n+6), n=4..50); # Robert Israel, Feb 02 2016

MATHEMATICA

Table[10 Binomial[2 n + 1, n - 4]/(n + 6), {n, 4, 28}] (* Michael De Vlieger, Feb 03 2016 *)

PROG

(PARI) a(n) = 10*binomial(2*n+1, n-4)/(n+6); \\ Michel Marcus, Feb 02 2016

(MAGMA) [10*Binomial(2*n+1, n-4)/(n+6): n in [4..35]]; // Vincenzo Librandi, Feb 03 2016

CROSSREFS

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, A001392.

Sequence in context: A073381 A092441 A022638 * A056280 A000453 A291231

Adjacent sequences:  A003516 A003517 A003518 * A003520 A003521 A003522

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 22 15:13 EST 2017. Contains 295089 sequences.