OFFSET
1,3
FORMULA
G.f. satisfies: A( x - A(x)^2 + A(x)^4 ) = x.
EXAMPLE
G.f.: A(x) = x + x^2 + 4*x^3 + 23*x^4 + 166*x^5 + 1380*x^6 + 12684*x^7 +...
Related series begin:
A(A(x)) = x + 2*x^2 + 10*x^3 + 67*x^4 + 538*x^5 + 4866*x^6 + 48000*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 174*x^5 + 1444*x^6 + 13224*x^7 +...
A(A(x))^4 = x^4 + 8*x^5 + 64*x^6 + 540*x^7 + 4856*x^8 + 46352*x^9 +...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 54*x^5 + 394*x^6 + 3276*x^7 + 29985*x^8 +...
A(x)^4 = x^4 + 4*x^5 + 22*x^6 + 144*x^7 + 1085*x^8 + 9100*x^9 +...
where the series reversion of the g.f. A(x) begins:
x - A(x)^2 + A(x)^4 = x - x^2 - 2*x^3 - 8*x^4 - 50*x^5 - 372*x^6 - 3132*x^7 -...
PROG
(PARI) {a(n)=local(A=x+x^2, B=x); for(i=1, n, B=subst(A, x, A+x*O(x^n)); A=x+B^2-B^4); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2+A^4+x*O(x^n))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 27 2012
STATUS
approved