

A162551


a(n) = 2 * C(2*n,n1).


16



0, 2, 8, 30, 112, 420, 1584, 6006, 22880, 87516, 335920, 1293292, 4992288, 19315400, 74884320, 290845350, 1131445440, 4407922860, 17194993200, 67156001220, 262564816800, 1027583214840, 4025232800160, 15780742227900, 61915399071552
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

Total length of all Dyck paths of length 2n.
a(n) equals the diagonal element A(n,n) of matrix A whose element A(i,j) = A(i1,j) + A(i,j1).  Carmine Suriano, May 10 2010
a(n) is also the number of solid (3 dimensions) standard Young tableaux of shape [[n,n],[1]].  Thotsaporn Thanatipanonda, Feb 27 2012
With offset = 1, a(n) is the total number of nodes over all binary trees with one child internal and one child external.  Geoffrey Critzer, Feb 23 2013
a(n) is the number of NorthEast paths from (0,0) to (n+1,n+1) that bounce off the diagonal y = x an odd number of times. Details can be found in Section 4.2 in Pan and Remmel's link.  Ran Pan, Feb 01 2016
a(n) is the number of NorthEast paths from (0,0) to (n+1,n+1) that cross the diagonal y = x an odd number of times. Details can be found in Section 4.3 in Pan and Remmel's link.  Ran Pan, Feb 01 2016


REFERENCES

R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley 1996, page 141.


LINKS



FORMULA

E.g.f.: exp(2*x)*2*(BesselI(1,2*x)).  Peter Luschny, Aug 26 2012
E.g.f.: 2*Q(0)  2, where Q(k) = 1  2*x/(k + 1  (k + 1)*(2*k + 3)/(2*k + 3  (k + 2)/Q(k+1))); (continued fraction).  Sergei N. Gladkovskii, Apr 28 2013


MATHEMATICA

nn=25; Drop[CoefficientList[Series[(12x)/(14x)^(1/2), {x, 0, nn}], x], 1] (* Geoffrey Critzer, Feb 23 2013 *)
Table[2Binomial[2n, n1], {n, 0, 30}] (* Harvey P. Dale, Oct 26 2016 *)


PROG

(Haskell)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



