|
|
A018224
|
|
a(n) = binomial(n, floor(n/2))^2 = A001405(n)^2.
|
|
7
|
|
|
1, 1, 4, 9, 36, 100, 400, 1225, 4900, 15876, 63504, 213444, 853776, 2944656, 11778624, 41409225, 165636900, 590976100, 2363904400, 8533694884, 34134779536, 124408576656, 497634306624, 1828114918084
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
a(n) is also the number of rooted two-vertex (or, dually, two-face) regular planar maps of valency n+1. - Valery A. Liskovets, Oct 19 2005
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n)=(-1)^n*E[(tr(A^4))^n]. - Andrew V. Sutherland, Apr 01 2008
|
|
LINKS
|
|
|
FORMULA
|
E.g.f.: BesselI(0, 2*x)*(BesselI(0, 2*x)+BesselI(1, 2*x)). - Vladeta Jovovic, Jun 12 2005
G.f. (1+1/(4*x))*hypergeom([1/2, 1/2],[1],16*x^2)-1/(4*x). - Mark van Hoeij, Oct 13 2009
a(n) = (n!/(floor(n/2)!*floor((n+1)/2)!))^2. - Peter Luschny, Apr 29 2014
a(n) = 4^n*[x^n]((1+x)*EllipticK(x)/(x*Pi/2)-1/x). - Peter Luschny, Oct 14 2015
a(n) ~ 4^n*((2*n+3)/(2*n+1))^((-1)^n/2)/((n+1)*Pi/2). - Peter Luschny, Oct 14 2015
a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)*C(k)*binomial(2*n-2*k,n-k) where C(k) are Catalan numbers (A000108), see Prodinger. - Michel Marcus, Nov 19 2019
Right hand side of the binomial sum identity (1/2)*Sum_{k = 0..n+1} (-1)^k*4^(n+1-k)*binomial(n+1,k)*binomial(n+k,k)*binomial(2*k,k) = a(n).
a(n) = (1/2)*4^(n+1) * hypergeom([n+1, -n-1, 1/2], [1, 1], 1).
P-recursive:
(2*n - 1)*(n + 1)^2*a(n) = 4*(2*n^2 - 1)*a(n-1) + 16*(2*n + 1)*(n - 1)^2*a(n-2) with a(0) = a(1) = 1. (End)
|
|
MAPLE
|
s := x -> (1+x)*EllipticK(x)/(x*Pi/2)-1/x:
seq(4^n*coeff(series(s(x), x, n+2), x, n), n=0..23); # Peter Luschny, Oct 14 2015
|
|
MATHEMATICA
|
(* Note that Mathematica uses a different definition of the EllipticK function. *)
CoefficientList[Series[(-Pi + (2 + 8 x) EllipticK[16 x^2])/(4 Pi x), {x, 0, 23}], x] (* Peter Luschny, Oct 14 2015 *)
Table[Binomial[n, Floor[n/2]]^2, {n, 0, 30}] (* Harvey P. Dale, Dec 02 2022 *)
|
|
PROG
|
(PARI) vector(50, n, n--; binomial(n, n\2)^2) \\ Altug Alkan, Oct 14 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|