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.
Cornelius 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].
Anders Claesson and Toufik Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001.
Milan Janjić, Two Enumerative Functions.
Cornelius 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., Vol. 6, No. 3 (2023), #P3.03.
Robert 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)
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.: hypergeom([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)
a(n) ~ 4^n / sqrt(Pi*n). - Amiram Eldar, Oct 11 2025
a(n) = binomial(n+3, 6) * Product_{1 <= i, j <= n-3} (i + j + 6)/(i + j + 5). - Peter Bala, Nov 17 2025
a(n) = A004310(n)*(n+4)/(n-3). - R. J. Mathar, Mar 16 2026
MAPLE
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
(SageMath) [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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Feb 18 2004
STATUS
approved
