A157309 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)) = o.g.f. of A157310 have zeros for every other coefficient after initial terms; g.f. of dual sequence A157308 satisfies the same condition.

1, 1, 0, -1, 0, 9, 0, -176, 0, 5693, 0, -272185, 0, 18043492, 0, -1587355800, 0, 179258676373, 0, -25305967691715, 0, 4370075849887361, 0, -906689353191842372, 0, 222613537277330398444, 0, -63850898347335510126988

Offset

0,6

Links

Table of n, a(n) for n=0..27.

Formula

Let F(x) = o.g.f. of A157310, 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 A157310 has zeros for every other term after initial [1,1,1].

Example

G.f.: A(x) = 1 + x - x^3 + 9*x^5 - 176*x^7 + 5693*x^9 -+...
RELATED FUNCTIONS.
If F(x) = A(x*F(x)) then F(x) = o.g.f. of A157310:
A157310 = [1,1,1,0,-3,0,38,0,-947,0,37394,0,-2120190,0,...];
has zeros for every other coefficient after initial terms.
...
O.g.f. A(x) has similar properties as o.g.f. of A157308:
A157308 = [1,1,-1,0,3,0,-38,0,947,0,-37394,0,2120190,0,...].

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))); Vec(x/serreverse(x*Ser(A)))[n+1]}

Crossrefs

Cf. A157308, A157310, A157307, A157304, A157305.

Sequence in context: A222516 A057403 A339488 * A215484 A177145 A178912

Adjacent sequences: A157306 A157307 A157308 * A157310 A157311 A157312

Keyword

sign

Author

Paul D. Hanna, Mar 11 2009

Status

approved