A003516 Binomial coefficients C(2n+1, n-2). (Formerly M4417)

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

Offset

2,2

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

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).

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

(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

Example

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

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 *)

Prog

(Magma) [Binomial(2*n+1, n-2): n in [2..25]]; // Vincenzo Librandi, Apr 13 2011
(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
(GAP) List([2..25], n-> Binomial(2*n+1, n-2)) # G. C. Greubel, Mar 21 2019

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.

Sequence in context: A026018 A085354 A051198 * A095931 A292486 A026856

Adjacent sequences: A003513 A003514 A003515 * A003517 A003518 A003519

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved