A003516 Binomial coefficients C(2n+1, n-2).

1, 7, 36, 165, 715, 3003, 12376, 50388, 203490, 817190, 3268760, 13037895, 51895935, 206253075, 818809200, 3247943160, 12875774670, 51021117810, 202112640600, 800472431850, 3169870830126, 12551759587422

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of royal paths (A006318) from (0,0) to (n,n) with exactly one diagonal step off the line y=x. - David Callan, Mar 25 2004

a(n) is the total number of DDUU's in all Dyck (n+2)-paths. - David Scambler, May 03 2013

Examples

			For n=4, C(2*4+1,4-2) = C(9,2) = 9*8/2 = 36, so a(4) = 36. - _Michael B. Porter_, Sep 10 2016
		

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.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 2..1000
• 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].
• Heidi Goodson, An Identity for Vertically Aligned Entries in Pascal's Triangle, arXiv:1901.08653 [math.CO], 2019.
• Milan Janjic, Two Enumerative Functions.
• Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022).
• Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.3. - From _N. J. A. Sloane_, Sep 16 2012
• 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 6 of triangle A100257.

Third unsigned column (s=2) of A113187. - Wolfdieter Lang, Oct 18 2012

Cf. triangle A114492 - Dyck paths with k DDUU's.

Cf. A001622, A115139.

Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

Programs

GAP

List([2..25], n-> Binomial(2*n+1, n-2)); # G. C. Greubel, Mar 21 2019

Magma

[Binomial(2*n+1,n-2): n in [2..25]]; // Vincenzo Librandi, Apr 13 2011

Mathematica

CoefficientList[ Series[ 32/(((Sqrt[1 - 4 x] + 1)^5)*Sqrt[1 - 4 x]), {x, 0, 25}], x] (* Robert G. Wilson v, Aug 08 2011 *)
Table[Binomial[2*n +1,n-2], {n,2,25}] (* G. C. Greubel, Jan 23 2017 *)

PARI

{a(n) = binomial(2*n+1, n-2)}; \\ G. C. Greubel, Mar 21 2019

Sage

[binomial(2*n+1, n-2) for n in (2..25)] # G. C. Greubel, Mar 21 2019

Formula

G.f.: 32*x^2/(sqrt(1-4*x)*(sqrt(1-4*x)+1)^5). - Marco A. Cisneros Guevara, Jul 18 2011

a(n) = Sum_{k=0..n-2} binomial(n+k+2,k). - Arkadiusz Wesolowski, Apr 02 2012

D-finite with recurrence (n+3)*(n-2)*a(n) = 2*n*(2*n+1)*a(n-1). - R. J. Mathar, Oct 13 2012

G.f.: x^2*c(x)^5/sqrt(1-4*x) = ((-1 + 2*x) + (1 - 3*x + x^2) * c(x))/(x^2*sqrt(1-4*x)), with c(x) the o.g.f. of the Catalan numbers A000108. See the W. Lang link under A115139 for powers of c. - Wolfdieter Lang, Sep 10 2016

a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - Ilya Gutkovskiy, Sep 10 2016

From Amiram Eldar, Jan 24 2022: (Start)

Sum_{n>=2} 1/a(n) = 4 - 14*Pi/(9*sqrt(3)).

Sum_{n>=2} (-1)^n/a(n) = 228*log(phi)/(5*sqrt(5)) - 134/15, where phi is the golden ratio (A001622). (End)

G.f.: 2F1([7/2,3],[6],4*x). - Karol A. Penson, Apr 24 2024

a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * sqrt(x)*(x^2 - 5*x + 5)/sqrt(4 - x). - Peter Bala, Oct 13 2024