OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
G.f. A(x) satisfies:
(1) A(x) = 2 * Series_Reversion( x - x*A(x) ) - x.
(2) A(x) = x * (1 + A(B(x))) / (1 - A(B(x))), where B(x) = (x + A(x))/2.
(3) A( (x + A(x))/2 ) = (A(x) - x) / (A(x) + x).
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 8*x^3 + 46*x^4 + 324*x^5 + 2608*x^6 + 23136*x^7 + 221370*x^8 + 2252872*x^9 + 24153284*x^10 + 270922880*x^11 + 3163154736*x^12 +...
such that A(x - x*A(x)) = x + x*A(x).
RELATED SERIES.
A(x - x*A(x)) = x + x^2 + 2*x^3 + 8*x^4 + 46*x^5 + 324*x^6 + 2608*x^7 +...
which equals x + x*A(x).
Series_Reversion( x - x*A(x) ) = x + x^2 + 4*x^3 + 23*x^4 + 162*x^5 + 1304*x^6 + 11568*x^7 + 110685*x^8 + 1126436*x^9 + 12076642*x^10 + 135461440*x^11 + 1581577368*x^12 +...
which equals (x + A(x))/2.
A( (x + A(x))/2 ) = x + 3*x^2 + 16*x^3 + 111*x^4 + 898*x^5 + 8068*x^6 + 78400*x^7 + 810875*x^8 + 8832804*x^9 + 100592970*x^10 + 1191393144*x^11 + 14616198024*x^12 +...
which equals (A(x) - x)/(A(x) + x).
PROG
(PARI) {a(n) = my(A=x); for(i=1, n, A = 2*serreverse( x - x*A +x*O(x^n) ) - x ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=x, B); for(i=1, n, B = (x + A)/2 +x*O(x^n); A = x*(1 + subst(A, x, B))/(1 - subst(A, x, B)) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 04 2016
STATUS
approved