

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.


3



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
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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: A222396 A222516 A057403 * A215484 A177145 A178912
Adjacent sequences: A157306 A157307 A157308 * A157310 A157311 A157312


KEYWORD

sign


AUTHOR

Paul D. Hanna, Mar 11 2009


STATUS

approved



