

A157310


G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x/F(x)) = o.g.f. of A157309 have zeros for every other coefficient after initial terms; g.f. of dual sequence A155585 satisfies the same condition.


4



1, 1, 1, 0, 3, 0, 38, 0, 947, 0, 37394, 0, 2120190, 0, 162980012, 0, 16330173251, 0, 2070201641498, 0, 324240251016266, 0, 61525045423103316, 0, 13913915097436287598, 0, 3698477457114061621492, 0
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OFFSET

0,5


COMMENTS

After initial 2 terms, reversing signs yields A157308.


LINKS

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


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))/(1x);
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))/(1x) 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

Cf. A157308, A157309, A155585, A157304, A157305, A122045 (Euler numbers).
Sequence in context: A276913 A012775 A157308 * A172396 A164806 A105751
Adjacent sequences: A157307 A157308 A157309 * A157311 A157312 A157313


KEYWORD

sign


AUTHOR

Paul D. Hanna, Mar 11 2009


STATUS

approved



