A107097 G.f. satisfies: A(A(x)) = A(x)/(1-x), so that the self-COMPOSE transform generates partial sums (A107098).

1, 1, 0, 1, -3, 13, -63, 339, -1982, 12429, -82827, 582589, -4303016, 33240205, -267697961, 2241725581, -19477340744, 175259713769, -1630583565434, 15663877511863, -155168272246709, 1583282220672515, -16623104947488348, 179409709469784087, -1988706708427161585

Offset

1,5

Links

Table of n, a(n) for n=1..25.

Formula

G.f. satisfies: A(x) = x + A(x)*Series_Reversion(A(x)).

Given g.f. A(x), let G(x) = Series_Reversion(A(x)), then G(x) satisfies:

(1) G(x) = 1 - x/A(x),

(2) G(x) = x - x*G(G(x)),

(3) -G(-x) is the g.f. of A030266, which shifts left under self-COMPOSE.

Example

G.f.: A(x) = x + x^2 + x^4 - 3*x^5 + 13*x^6 - 63*x^7 + 339*x^8 -+...
If G(x) = series reversion of g.f. A(x) so that A(G(x)) = x, then G(x) begins:
G(x) = x - x^2 + 2*x^3 - 6*x^4 + 23*x^5 - 104*x^6 + 531*x^7 - 2982*x^8 -+...
Compare the functional inverse, G(x), to the arithmetic inverse x/A(x):
x/A(x) = 1 - x + x^2 - 2*x^3 + 6*x^4 - 23*x^5 + 104*x^6 - 531*x^7 + 2982*x^8 -+...

Prog

(PARI) {a(n)=local(A, B, F); if(n<1, 0, F=x+2*x^2-3*x^3+x*O(x^n); A=F; for(j=0, n, for(i=0, j, B=serreverse(A); A=(A+subst(B, x, A/(1-x)))/2); A=round(A)); polcoeff(A, n, x))}
(PARI) /* A(x) = x + A(x)*Series_Reversion(A(x)): */
{a(n)=local(A=x+x^2); for(i=1, n, A=x+A*serreverse(A+x*O(x^n))); polcoeff(A, n)}

Crossrefs

Cf. A107098.

Sequence in context: A367061 A192729 A284716 * A202837 A370396 A180111

Adjacent sequences: A107094 A107095 A107096 * A107098 A107099 A107100

Keyword

sign

Author

Paul D. Hanna, May 12 2005, Jul 23 2011

Extensions

Initial zero removed and offset changed to 1 by Paul D. Hanna, Jul 23 2011

Status

approved