OFFSET
0,4
FORMULA
EXAMPLE
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,...].
PROG
(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]}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 28 2009
STATUS
approved