OFFSET
0,5
COMMENTS
After initial 2 terms, reversing signs yields A157308.
FORMULA
Let F(x) = A(x/F(x)) = o.g.f. of A157309, then F(x) satisfies:
A(x) = Series_Reversion(x/F(x))/x;
A(x) = F(x*A(x));
F(x) = A(x/F(x));
where A157309 has zeros for every other term after initial [1,1].
...
Let G(x) = o.g.f. of A122045, then o.g.f. A(x) satisfies:
A(x) = 2+x - x/Series_Reversion(x*G(x));
A(x) = 2+x - G( x/(2+x - A(x)) );
G(x) = (2 - A(x*G(x))/(1-x);
where A122045 is the Euler numbers.
...
Let H(x) = o.g.f. of A155585, then o.g.f. A(x) satisfies:
A(x) = 2(1+x) - x/Series_Reversion(x*H(x));
A(x) = 2 - H( -x/(2 - A(x)) );
A(x) = H(-x/A(x));
H(x) = A(x*H(-x));
where A155585 is defined by e.g.f. exp(x)/cosh(x).
...
O.g.f.: A(x) = 2*(1+x) - B(x) where B(x) = g.f. of A157308.
EXAMPLE
G.f.: A(x) = 1 + x + x^2 - 3*x^4 + 38*x^6 - 947*x^8 + 37394*x^10 -+...
RELATED FUNCTIONS.
If F(x) = A(x/F(x)) then F(x) = o.g.f. of A157309:
A157309 = [1,1,0,-1,0,9,0,-176,0,5693,0,-272185,0,...];
...
If G(x) = (2 - A(x*G(x))/(1-x) then G(x) = o.g.f. of A122045:
A122045 = [1,0,-1,0,5,0,-61,0,1385,0,-50521,0,2702765,0,...];
...
Let H(x) = A(x*H(-x)) = o.g.f. of A155585:
A155585 = [1,1,0,-2,0,16,0,-272,0,7936,0,-353792,0,...];
...
PROG
(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==1, A=concat(A, 0); ); if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); A[n+1]}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Mar 11 2009
STATUS
approved