|
|
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].
|
|
FORMULA
|
(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
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
|
(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
|
Cf. triangle A114492 - Dyck paths with k DDUU's.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|