OFFSET
0,3
COMMENTS
Old name was: Central coefficients of number triangle A119326.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (1/sqrt(1-4x)+1/sqrt(1+4x^2))/2.
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)^2.
a(n) = C(2n,n)/2+sin(Pi*(n+1)/2)*C(n,n/2)/2.
a(n) = A119326(2n,n).
D-finite with recurrence n*(n-1)*(10*n-29)*a(n) +2*(n-1)*(5*n^2-74*n+164)*a(n-1) +4*(-40*n^3+310*n^2 -744*n+559)*a(n-2) +8*(n-2)*(5*n^2-74*n+164)*a(n-3) -16*(25*n-42)*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 05 2012
a(n) = hypergeom([(1-n)/2, (1-n)/2, -n/2, -n/2], [1/2, 1/2, 1], 1). - Vladimir Reshetnikov, Oct 04 2016
a(n) = A282011(2n,n). - Alois P. Heinz, Feb 04 2017
EXAMPLE
a(3) = 10: {1,2,3}, {1,2,5}, {1,3,4}, {1,3,6}, {1,4,5}, {1,5,6}, {2,3,5}, {2,4,6}, {3,4,5}, {3,5,6}. - Alois P. Heinz, Feb 04 2017
MAPLE
a:= proc(n) option remember; `if`(n<3, 1+n*(n-1)/2,
((4*n-10)*(5*n^2-10*n+4)*(a(n-1)+4*(n-2)*a(n-3)
/(n-1))/(5*n^2-20*n+19)-4*(n-1)*a(n-2))/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Aug 26 2018
MATHEMATICA
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1/2, 1/2, 1}, 1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 04 2016 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 16 2006
EXTENSIONS
New name from Alois P. Heinz, Feb 04 2017
STATUS
approved