A157302 G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)^2) have zeros for every other coefficient after initial terms; g.f. of dual sequence A157305 satisfies the same condition.

1, 1, 0, -5, 0, 183, 0, -14352, 0, 1857199, 0, -355082433, 0, 94134281460, 0, -33120720127500, 0, 14959943533260783, 0, -8447188671812872887, 0, 5834800994047642310223, 0, -4842259038722174600622240, 0

Offset: 0

Author

Paul D. Hanna, Feb 28 2009

Keywords

• sign

Examples

			G.f.: A(x) = 1 + x - 5*x^3 + 183*x^5 - 14352*x^7 + 1857199*x^9 -+...
...
Let F(x) = A(x*F(x)^2) so that A(x) = F(x/A(x)^2) then
F(x) = 1 + x + 2*x^2 - 26*x^4 + 1378*x^6 - 141202*x^8 +-...
has alternating zeros in the coefficients (cf. A157304):
[1,1,2,0,-26,0,1378,0,-141202,0,22716418,0,-5218302090,0,...].
...
COEFFICIENTS IN ODD POWERS OF G.F. A(x).
A^1: [(1),1,0,-5,0,183,0,-14352,0,1857199,0,...];
A^3: [1,(3),3,-14,-30,534,1173,-42432,-91602,5522926,...];
A^5: [1,5,(10),-15,-95,766,3810,-65545,-300930,8800450,...];
A^7: [1,7,21,(0),-175,777,7518,-79148,-610554,11321338,...];
A^9: [1,9,36,39,(-234),513,11640,-79866,-990603,...];
A^11:[1,11,55,110,-220,(0),15367,-66132,-1402005,...];
A^13:[1,13,78,221,-65,-624,(17914),-38571,-1801215,...];
A^15:[1,15,105,380,315,-1077,18760,(0),-2145855,...];
A^17:[1,17,136,595,1020,-901,17952,45084,(-2400434),...];
A^19:[1,19,171,874,2166,570,16473,91656,-2541060,(0),...];
...
When scaled, the coefficients shown above in parenthesis
forms the coefficients of the function F(x) = A(x*F(x)^2):
F: [1,3/3,10/5,0,-234/9,0,17914/13,0,-2400434/17,0,...].
		

Crossrefs

Cf. A157303, A157304, A157305 (dual), A157306, A157307.

Programs

PARI

{a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(serreverse(x/Ser(A)))[ #A], t, 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]}

Formula

For n>=2, [x^(2n-1)] A(x)^(4n-1) = 0.

G.f. satisfies: A(x) = F(x/A(x)^2) where F(x) = A(x*F(x)^2) = sqrt(Series_Reversion(x/A(x)^2)/x) = g.f. of A157304.

G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) = Series_Reversion(x/A(x))/x = g.f. of A157303.