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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 (list; graph; refs; listen; history; text; internal format)
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))/(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

Cf. A157308, A157309, A155585, A157304, A157305, A122045 (Euler numbers).

Sequence in context: A012775 A157308 * A172396 A164806 A105751 A177698

Adjacent sequences:  A157307 A157308 A157309 * A157311 A157312 A157313

KEYWORD

sign

AUTHOR

Paul D. Hanna, Mar 11 2009

STATUS

approved

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Last modified August 19 23:35 EDT 2017. Contains 290821 sequences.