OFFSET
0,3
COMMENTS
Compare to: G(x) = Series_Reversion( x - Series_Reversion(x*G(x)) * x*G(x) )/x, which is satisfied by the g.f. G(x) = 1 + x*G(x) * G(x*G(x)) of A030266 with offset 0.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Series_Reversion( x - Series_Reversion(x/A(x)) * x/A(x) )/x.
(2) A(x) = x/Series_Reversion( (x - Series_Reversion(x*A(x))) * A(x)/x ).
Given B(x) = A(x/B(x)) and C(x) = A(x*C(x)), then:
(3.a) A(x) = B(x*A(x)) and A(x) = C(x/A(x)),
(3.b) B(x) = x/Series_Reversion(x*A(x)),
(3.c) C(x) = Series_Reversion(x/A(x))/x,
(3.d) B(x) = A(x)/(A(x) - x*C(x)),
(3.e) C(x) = A(x)*(1 - 1/B(x))/x.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 149*x^5 + 1024*x^6 +...
Let B(x) = A(x/B(x)) and C(x) = A(x*C(x)), where B(x) and C(x) begin:
B(x) = 1 + x + x^2 + 2*x^3 + 9*x^4 + 53*x^5 + 354*x^6 + 2651*x^7 + 21951*x^8 + 197666*x^9 + 1911091*x^10 + 19665622*x^11 + 214060860*x^12 +...
C(x) = 1 + x + 3*x^2 + 13*x^3 + 71*x^4 + 460*x^5 + 3399*x^6 + 27867*x^7 + 248789*x^8 + 2388199*x^9 + 24432778*x^10 + 264682253*x^11 + 3021086129*x^12 +...
then A(x) = B(x) * (A(x) - x*C(x)).
PROG
(PARI) {a(n)=local(A=1); for(i=1, n, A=serreverse(x-x/A*serreverse(x/(A +x^2*O(x^n))))/x); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 19 2014
STATUS
approved