OFFSET
1,2
COMMENTS
a(38) = -19599187894639322176080463718044944 is the first negative term.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = A( A(x)*(1 - 2*A(x))^2 ).
(2) x = A(A( x*(1 - 2*x)^2 )).
(3) x = B(x)*(1 - 2*B(x))^2 where B(x) = A(A(x)) (A369510).
(4) A(x) = Series_Reversion( A(x)*(1 - 2*A(x))^2 ).
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 10*x^3 + 66*x^4 + 498*x^5 + 4056*x^6 + 34644*x^7 + 305310*x^8 + 2749110*x^9 + 25142172*x^10 + 232728588*x^11 + 2176116348*x^12 + ...
where A( A(x)*(1 - 2*A(x))^2 ) = x.
RELATED SERIES.
Let B(x) = A(A(x)), then B(x)/x is the g.f. of A369510:
B(x) = x + 4*x^2 + 28*x^3 + 240*x^4 + 2288*x^5 + 23296*x^6 + 248064*x^7 + 2728704*x^8 + 30764800*x^9 + ... + 2^(n-1)*binomial(3*n-2,n-1)/n * x^n + ...
where B(x)*(1 - 2*B(x))^2 = x.
Let R(x) be the series reversion, R(A(x)) = x, then
R(x) = A(x)*(1 - 2*A(x))^2 = x - 2*x^2 - 2*x^3 - 6*x^4 - 22*x^5 - 80*x^6 - 228*x^7 - 18*x^8 + 6694*x^9 + ... + (-1)^(n-1)*A097090(n)*x^n + ...
where R(R(x)) = x*(1 - 2*x)^2.
PROG
(PARI) {a(n) = my(A=x); for(k=2, n+1, A=truncate(A); A = (A + serreverse( A*(1 - 2*A)^2 +x*O(x^k)))/2 ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 27 2024
STATUS
approved