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A002696 Binomial coefficients C(2n,n-3).
(Formerly M4532 N1921)
7
1, 8, 45, 220, 1001, 4368, 18564, 77520, 319770, 1307504, 5311735, 21474180, 86493225, 347373600, 1391975640, 5567902560, 22239974430, 88732378800, 353697121050, 1408831480056, 5608233007146, 22314239266528, 88749815264600 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=3. - Herbert Kociemba, May 23 2004

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.

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

Robert Israel, Table of n, a(n) for n = 3..1497

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

A. Claesson and T. Mansour, Counting patterns of type (1,2) or (2,1), arXiv:math/0110036 [math.CO], 2001.

Milan Janjic, Two Enumerative Functions

C. Lanczos, Applied Analysis (Annotated scans of selected pages)

R. Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

Hermann Stamm-Wilbrandt, Compute C(2n, n-k) based on C(n,...) animation

Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

FORMULA

G.f.: (1-sqrt(1-4*z))^6/(64*z^3*sqrt(1-4*z)). - Emeric Deutsch, Jan 28 2004

a(n) = Sum_{k=0..n} C(n, k)*C(n, k+3). - Hermann Stamm-Wilbrandt, Aug 17 2015

From Robert Israel, Aug 19 2015: (Start)

(n-2)*(n+4)*a(n+1) = (2*n+2)*(2*n+1)*a(n).

E.g.f.: I_3(2*x)*exp(2*x) where I_3 is a modified Bessel function. (End)

MAPLE

A002696:=n->binomial(2*n, n-3): seq(A002696(n), n=3..30); # Wesley Ivan Hurt, Aug 19 2015

MATHEMATICA

CoefficientList[Series[64/(((Sqrt[1-4x] +1)^6)*Sqrt[1-4x]), {x, 0, 30}], x] (* Robert G. Wilson v, Aug 08 2011 *)

PROG

(MAGMA) [ Binomial(2*n, n-3): n in [3..30] ]; // Vincenzo Librandi, Apr 13 2011

(PARI) a(n)=binomial(n+n, n-3) \\ Charles R Greathouse IV, Aug 08 2011

(Sage) [binomial(2*n, n-3) for n in (3..30)] # G. C. Greubel, Mar 21 2019

(GAP) List([3..30], n-> Binomial(2*n, n-3)) # G. C. Greubel, Mar 21 2019

CROSSREFS

Diagonal 7 of triangle A100257.

Column k=1 of A263776.

Sequence in context: A055222 A273305 A026015 * A016208 A216540 A026852

Adjacent sequences:  A002693 A002694 A002695 * A002697 A002698 A002699

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Emeric Deutsch, Feb 18 2004

STATUS

approved

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Last modified October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)