OFFSET
1,2
COMMENTS
From Wolfdieter Lang, Jan 02 2012: (Start)
a(n) = A024458(2*n-1), n>=1 (bisection, odd arguments).
chate(n):=a(n+1), n>=0, is the even part of the bisection of the half-convolution of the sequence A000045(n+1), n>=0, with itself. See a comment on A201204 for the definition of half-convolution. There one finds also the rule for the o.g.f.s of the bisection. Here the o.g.f. of the sequence chate(n), n>=0, is Chate(x):= (Ce(x)+U2(x))/2 with Ce(x)=(1-x+x^2)/(1-3*x+x^2)^2, the o.g.f. of A054444(n), and
U2(x)=(1-x)/((1+x)*(1-3*x+x^2)), the o.g.f. of A007598(n+1), n>=0. This results (after multiplying with x) in the o.g.f. given below in the formula section. It is equivalent to the explicit formula given there, as can be seen after a partial fraction decomposition of the o.g.f.
(End)
FORMULA
a(n) = (1/5)[n*F(2n+2) - n*F(2n-2) + F(2n-1) - (-1)^n], F(n)=A000045(n).
O.g.f.: x*(1-2*x+2*x^2)/((1-3*x+x^2)^2*(1+x)). See the comment above. - Wolfdieter Lang, Jan 02 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved