login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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, 352870329957600, 1402659561581460 (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
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.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022).
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.
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)
From Amiram Eldar, Aug 27 2022: (Start)
Sum_{n>=3} 1/a(n) = 3/4 + 2*Pi/(9*sqrt(3)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 444*log(phi)/(5*sqrt(5)) - 1093/60, where phi is the golden ratio (A001622). (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.
Cf. A001622.
Sequence in context: A055222 A273305 A026015 * A016208 A216540 A026852
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Feb 18 2004
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)