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A004310 Binomial coefficient C(2n,n-4). 3
1, 10, 66, 364, 1820, 8568, 38760, 170544, 735471, 3124550, 13123110, 54627300, 225792840, 927983760, 3796297200, 15471286560, 62852101650, 254661927156, 1029530696964, 4154246671960, 16735679449896 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,2

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=4. - Herbert Kociemba, May 23 2004

LINKS

Table of n, a(n) for n=4..24.

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], p 828

Milan Janjic, Two Enumerative Functions

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From N. J. A. Sloane, Feb 13 2013

M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5

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

FORMULA

-(n-4)*(n+4)*a(n) +2*n*(2*n-1)*a(n-1)=0. - R. J. Mathar, Dec 22 2013

G.f.: x*(1/(sqrt(1-4*x)*x)-(1-sqrt(1-4*x))/(2*x^2))/((1-sqrt(1-4*x))/(2*x)-1)^5-(1/x^4-6/x^3+10/x^2-4/x). - Vladimir Kruchinin, Aug 11 2015

a(n) = Sum_{k=0..n} C(n, k)*C(n, k+4). - Hermann Stamm-Wilbrandt, Aug 17 2015

E.g.f.: BesselI(4,2*x)*exp(2*x). - Ilya Gutkovskiy, Jun 27 2019

PROG

(MAGMA) [ Binomial(2*n, n-4): n in [4..150] ]; // Vincenzo Librandi, Apr 13 2011

(PARI) first(m)=vector(m, i, binomial(2*(i+3), i-1)) \\ Anders Hellström, Aug 17 2015

CROSSREFS

Diagonal 9 of triangle A100257.

Sequence in context: A074362 A080421 A320817 * A026853 A177452 A033504

Adjacent sequences:  A004307 A004308 A004309 * A004311 A004312 A004313

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 13 09:12 EDT 2021. Contains 342935 sequences. (Running on oeis4.)