A002696 Binomial coefficients C(2n,n-3).

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

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

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)
• 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.
• 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).

Crossrefs

Diagonal 7 of triangle A100257.

Column k=1 of A263776.

Cf. A001622.

Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A030053 - A030056, A004310 - A004318.

Programs

GAP

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

Magma

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

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 *)

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

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)

G.f.: 2F1([7/2,4],[7],4*x). - Karol A. Penson, Apr 24 2024

From Peter Bala, Oct 13 2024: (Start)

a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * (x^3 - 6*x^2 + 9*x - 2)/sqrt(x*(4 - x)).

G.f: x^3 * B(x) * C(x)^6, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

Extensions

More terms from Emeric Deutsch, Feb 18 2004