login
A004311
Binomial coefficient C(2n,n-5).
4
1, 12, 91, 560, 3060, 15504, 74613, 346104, 1562275, 6906900, 30045015, 129024480, 548354040, 2310789600, 9669554100, 40225345056, 166509721602, 686353797976, 2818953098830, 11541847896480, 47129212243960, 191991813933920, 780512175396135, 3167295784216200
OFFSET
5,2
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=5. - Herbert Kociemba, May 24 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.
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].
Milan Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013
Milan Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014), Article 14.3.5.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
FORMULA
a(n) = Sum{k=0..n} C(n, k)*C(n, k+5). - Hermann Stamm-Wilbrandt, Aug 17 2015
-(n-5)*(n+5)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Jan 24 2018
E.g.f.: BesselI(5,2*x)*exp(2*x). - Ilya Gutkovskiy, Jun 27 2019
From Amiram Eldar, Aug 27 2022: (Start)
Sum_{n>=5} 1/a(n) = 6169/840 - 31*Pi/(9*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 5254*log(phi)/(5*sqrt(5)) - 63059/280, where phi is the golden ratio (A001622). (End)
G.f.: 2F1([11/2,6],[11],4*x). - Karol A. Penson, Apr 24 2024
MATHEMATICA
Table[Binomial[2*n, n-5], {n, 5, 30}] (* Amiram Eldar, Aug 27 2022 *)
PROG
(Magma) [ Binomial(2*n, n-5): n in [5..150] ]; // Vincenzo Librandi, Apr 13 2011
(PARI) first(m)=vector(m, i, binomial(2*(i+4), i-1)) \\ Anders Hellström, Aug 17 2015
CROSSREFS
Diagonal 11 of triangle A100257.
Cf. A001622.
Sequence in context: A114860 A001502 A001503 * A160869 A026074 A339715
KEYWORD
nonn,easy
STATUS
approved