OFFSET
0,2
COMMENTS
Number of 4-ary words of length 2n in which the combined count of 0's and 1's is not greater than n. - David Scambler, Aug 13 2012
REFERENCES
The right-hand side of a binomial coefficient identity in H. W. Gould, Combinatorial Identities, Morgantown, 1972, (3.97), page 33.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..825
H. W. Gould, ed. J. Quaintance, Combinatorial Identities, May 2010, (7.5) p. 34.
FORMULA
a(n) + A055787(n) = 2^(4n). - David Scambler, Aug 13 2012
n*a(n) +8*(3-4*n)*a(n-1) +128*(2*n-3)*a(n-2)=0. - R. J. Mathar, Aug 13 2012 [Proof by inserting the binomial expression at all three places, converting everything to Gamma-functions, and let Maple 16 simplify to zero. Robert Israel, Aug 13 2012]
a(n) = 8/n*((2*n-1)*a(n-1)+2^(4*n-5)). Using this formula we can express a(n) and a(n-1) via a(n-2) and easily obtain the hypergeometric recurrence above. - Vladimir Shevelev, Aug 13 2012
a(n) = 2^(4*n-1)*((Gamma(n+1/2))/(sqrt(Pi)*Gamma(n+1))+1). - Alexander R. Povolotsky, Aug 13 2012
a(n) = Sum_{k=0..n}(C(4*k, 2*k) * C(4*n-4*k, 2*n-2*k)). (LHS of Gould's identity).
a(n) = Sum_{k=0..n}(2^(2*n) * C(2*n, k)). - David Scambler, Aug 13 2012
MATHEMATICA
Table[2^(2n-1) Binomial[2n, n]+2^(4n-1), {n, 0, 30}] (* Harvey P. Dale, Mar 14 2013 *)
PROG
(PARI) for(n=0, 25, print1(2^(2*n-1)*binomial(2*n, n) + 2^(4*n-1), ", ")) \\ G. C. Greubel, Feb 16 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 15 2000
STATUS
approved