A157303 G.f. A(x) satisfies the condition that both G(x) = A(x/G(x)) and H(x) = A(x*H(x)) have zeros for every other coefficient after initial terms; g.f. of dual sequence A157306 satisfies the same condition.

1, 1, 1, -4, -19, 134, 1074, -10158, -110067, 1302086, 17451662, -248857456, -3948994550, 66104803660, 1210719480268, -23304935437410, -484165206834051, 10541640152174406, 245149591293286518, -5958110923842801192

Offset

0,4

Links

Table of n, a(n) for n=0..19.

Formula

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

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

G.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = x/Series_Reversion(x*A(x)) = g.f. of A157302.

G.f. satisfies: A(x) = H(x/A(x)) where H(x) = A(x*H(x)) = Series_Reversion(x/A(x))/x = g.f. of A157304.

Example

G.f.: A(x) = 1 + x + x^2 - 4*x^3 - 19*x^4 + 134*x^5 + 1074*x^6 --++...
...
Let G(x) = A(x/G(x)) so that A(x) = G(x*A(x)) then
G(x) = 1 + x - 5*x^3 + 183*x^5 - 14352*x^7 + 1857199*x^9 -+...
has alternating zeros in the coefficients (cf. A157302):
[1,1,0,-5,0,183,0,-14352,0,1857199,0,-355082433,0,94134281460,0,...]
...
Let H(x) = A(x*H(x)) so that A(x) = H(x/A(x)) then
H(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,...].
...
ZERO COEFFICIENTS IN POWERS OF G.F. A(x).
Even powers A(x)^(2n) yield zeros at odd positions 2n-1 for n>=2:
A^4: [1,4,10, 0, -105,228,5442,-24048,-535293,3588828,...];
A^6: [1,6,21,26,-144, 0, 8415,-17802,-849771,3596382,...];
A^8: [1,8,36,80,-110,-384,10608, 0, -1143105,2366472,...];
A^10:[1,10,55,170,65,-718,11580,26520,-1381080, 0, ...];
...
Odd negative powers 1/A(x)^(2n-1) yield zeros at even positions 2n for n>=1:
A^-1: [1,-1, 0, 5,10,-168,-762,12297,87222,-1524622,...];
A^-3: [1,-3,3,14, 0, -549,-1173,41184,180558,-5088382,...];
A^-5: [1,-5,10,15,-45,-891, 0, 71760,152670,-9042640,...];
A^-7: [1,-7,21,0,-105,-1092,2604,99042, 0, -13000393,...]; ...

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^2/serreverse(x*Ser(A)))[ #A], t, 0))); A[n+1]}

Crossrefs

Cf. A157302, A157304, A157305, A157306 (dual), A157307.

Sequence in context: A127060 A134147 A220066 * A007788 A259354 A215851

Adjacent sequences: A157300 A157301 A157302 * A157304 A157305 A157306

Keyword

sign

Author

Paul D. Hanna, Feb 28 2009

Status

approved