|
| |
|
|
A350433
|
|
G.f. A(x) satisfies: R(x^2*A(x)) = x^3 - x^4, where R(A(x)) = x.
|
|
7
|
|
|
|
1, -1, 0, -1, 2, -1, 0, 0, 0, -1, 4, -6, 6, -11, 20, -21, 16, -17, 20, -15, 6, -1, 0, 0, 0, 0, 0, -1, 10, -45, 124, -254, 472, -876, 1512, -2289, 3184, -4369, 5880, -7491, 9240, -11697, 15640, -22003, 31916, -46450, 67104, -96081, 133976, -178209, 226608
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,5
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x, and F(n+1) = 1 - (1 - F(n))^3 * F(n) for n > 0.
|
|
|
EXAMPLE
|
G.f.: A(x) = x - x^2 - x^4 + 2*x^5 - x^6 - x^10 + 4*x^11 - 6*x^12 + 6*x^13 - 11*x^14 + 20*x^15 - 21*x^16 + 16*x^17 + ...
Let R(x) be the series reversion of A(x),
R(x) = x + x^2 + 2*x^3 + 6*x^4 + 18*x^5 + 57*x^6 + 192*x^7 + 666*x^8 + 2362*x^9 + 8548*x^10 + ... + A273203(n)*x^n + ...
then R(x) and g.f. A(x) satisfy:
(1) R(A(x)) = x,
(2) R(x^2*A(x)) = x^3 - x^4.
GENERATING METHOD.
Define F(n), a polynomial in x of order 4^(n-1), by the following recurrence:
F(0) = x,
F(1) = (1 - x),
F(2) = (1 - x^3 * (1-x)),
F(3) = (1 - x^9 * (1-x)^3 * F(2)),
F(4) = (1 - x^27 * (1-x)^9 * F(2)^3 * F(3)),
F(5) = (1 - x^81 * (1-x)^27 * F(2)^9 * F(3)^3 * F(4)),
...
F(n+1) = 1 - (1 - F(n))^3 * F(n)
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
that is,
A(x) = x * (1-x) * (1 - x^3*(1-x)) * (1 - x^9*(1-x)^3*(1 - x^3*(1-x))) * (1 - x^27*(1-x)^9*(1 - x^3*(1-x))^3*(1 - x^9*(1-x)^3*(1 - x^3*(1-x)))) * ...
SPECIFIC VALUES.
The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.
A(1/2) = (1/2) * (1/2) * (15/2^4) * (65521/2^16) * (18446744073488418241/2^64) * ... = 0.23432135581689318269048583...
A(2/3) = (2/3) * (1/3) * (73/3^4) * (43009345/3^16) * (3433683818046866504964267490561/3^64) * ... = 0.20010045690402248041761236...
A(1/3) = (1/3) * (2/3) * (79/3^4) * (43046089/3^16) * (3433683820292501618276703760129/3^64) * ... = 0.21673207172549998378147271...
The first relative maximum value of A(x) is given by
A(0.467248308756732593319037623545720...) = 0.235363377004699385202253369020528...
|
|
|
PROG
|
(PARI) /* Using Functional Equation in Definition */
{a(n) = my(A=[1, -1], B); for(i=1, n, A = concat(A, 0);
R = serreverse(x*Ser(A));
A[#A] = polcoeff( x^3 - x^4 - subst(R, x, x^3*Ser(A)), #A+2) ); H=A; A[n]}
for(n=1, 60, print1(a(n), ", "))
(PARI) /* Using Infinite Product Formula */
{F(n) = my(G=x); if(n==0, G=x, if(n==1, G=1-x, G = 1 - (1 - F(n-1))^3*F(n-1) )); G}
{a(n) = my(A = prod(k=0, round(log(n)/log(3))+1, F(k) +x*O(x^n))); polcoeff(A, n)}
for(n=1, 60, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
