OFFSET
0,3
COMMENTS
Average of binomial and inverse binomial transforms of the Catalan numbers A000108.
FORMULA
G.f.: (2-sqrt((1-3x)/(1+x))-sqrt((1-5x)/(1-x)))/(4x);
a(n)=sum{k=0..floor(n/2), binomial(n, 2k)C(n-2k)};
a(n)=sum{k=0..n, binomial(n, k)C(k)(1+(-1)^(n-k))/2}.
Conjecture: -(n-1)*(n+1)*a(n) +2*(5*n^2-9*n+1)*a(n-1) +2*(-15*n^2+58*n-49)*a(n-2) +2*(10*n^2-76*n+123)*a(n-3) +(31*n-55)*(n-3)*a(n-4) -30*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 08 2016
Conjecture: +(3*n-10)*(n-1)*(n+1)*a(n) +2*(-12*n^3+58*n^2-67*n+10)*a(n-1) +2*(21*n^3-136*n^2+289*n-196)*a(n-2) +2*(n-2)*(12*n^2-46*n+27)*a(n-3) -15*(n-2)*(n-3)*(3*n-7)*a(n-4)=0. - R. J. Mathar, Jun 08 2016
a(n) ~ 5^(n + 3/2) / (16 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 30 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jan 04 2005
STATUS
approved