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A003516 Binomial coefficients C(2n+1, n-2).
(Formerly M4417)
11
1, 7, 36, 165, 715, 3003, 12376, 50388, 203490, 817190, 3268760, 13037895, 51895935, 206253075, 818809200, 3247943160, 12875774670, 51021117810, 202112640600, 800472431850, 3169870830126, 12551759587422 (list; graph; refs; listen; history; text; internal format)
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
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.
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)
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.
Sequence in context: A026018 A085354 A051198 * A095931 A292486 A026856
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 18 22:50 EDT 2024. Contains 370951 sequences. (Running on oeis4.)