OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
G.f. A(x) satisfies: A( A(x^2 - 2*x^3) / x ) = x.
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 6*x^3 + 28*x^4 + 150*x^5 + 848*x^6 + 4988*x^7 + 30320*x^8 + 189030*x^9 + 1201792*x^10 +...
where A( A(x)^2 - 2*A(x)^3 ) = x*A(x).
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 80*x^5 + 448*x^6 + 2632*x^7 + 15952*x^8 + 99168*x^9 + 629184*x^10 + 4057272*x^11 + 26511544*x^12 +...
A(x)^3 = x^3 + 6*x^4 + 30*x^5 + 164*x^6 + 966*x^7 + 5904*x^8 + 36924*x^9 + 235248*x^10 + 1522086*x^11 + 9974080*x^12 + 66055800*x^13 +...
A(x)^2 - 2*A(x)^3 = x^2 + 2*x^3 + 4*x^4 + 20*x^5 + 120*x^6 + 700*x^7 + 4144*x^8 + 25320*x^9 + 158688*x^10 + 1013100*x^11 + 6563384*x^12 +...
A(x^2 - 2*x^3) = x^2 - 2*x^3 + 2*x^4 - 8*x^5 + 14*x^6 - 36*x^7 + 100*x^8 - 272*x^9 + 822*x^10 - 2396*x^11 + 7296*x^12 - 22176*x^13 + 67868*x^14 +...
Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
B(x) = x - 2*x^2 + 2*x^3 - 8*x^4 + 14*x^5 - 36*x^6 + 100*x^7 - 272*x^8 + 822*x^9 - 2396*x^10 + 7296*x^11 - 22176*x^12 + 67868*x^13 +...
where B(x) = A(x^2 - 2*x^3)/x,
also, B( x*B(x) ) = x^2 - 2*x^3.
PROG
(PARI) {a(n) = my(A=[1], F); for(i=1, n, A = concat(A, 0); F = x*Ser(A); A[#A] = -Vec(subst(F, x, F^2 - 2*F^3))[#A]); A[n]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 12 2016
STATUS
approved