A350434 G.f. A(x) satisfies: A(x^3*R(x)) = x^4 - x^5 where A(R(x)) = x.

1, 1, 2, 5, 15, 47, 153, 513, 1765, 6193, 22071, 79664, 290636, 1070030, 3970554, 14834370, 55755406, 210671102, 799783375, 3049139589, 11669146635, 44812839601, 172636642278, 666981050965, 2583695305266, 10032884823527, 39046925631765, 152283522642736, 595059178361120

Offset

1,3

Links

Table of n, a(n) for n=1..29.

Formula

G.f.: A(x) = Series_Reversion( Product_{n>=0} F(n) ), where F(0) = x, F(1) = 1-x, and F(n+1) = 1 - (1 - F(n))^4 * F(n) for n > 0.

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 47*x^6 + 153*x^7 + 513*x^8 + 1765*x^9 + 6193*x^10 + 22071*x^11 + ...
Let R(x) be the series reversion of A(x),
R(x) = x - x^2 - x^5 + 2*x^6 - x^7 - x^17 + 5*x^18 - 10*x^19 + 10*x^20 - 3*x^21 - 11*x^22 + 30*x^23 - 40*x^24 + 29*x^25 + ...
then R(x) and g.f. A(x) satisfy:
(1) A(R(x)) = x,
(2) A(x^3*R(x)) = x^4 - x^5.
GENERATING METHOD.
Define F(n), a polynomial in x of order 5^(n-1), by the following recurrence:
F(0) = x,
F(1) = (1 - x),
F(2) = (1 - x^4 * (1-x)),
F(3) = (1 - x^16 * (1-x)^4 * F(2)),
F(4) = (1 - x^64 * (1-x)^16 * F(2)^4 * F(3)),
F(5) = (1 - x^256 * (1-x)^64 * F(2)^16 * F(3)^4 * F(4)),
...
F(n+1) = 1 - (1 - F(n))^4 * F(n)
...
Then the series reversion R(x) equals the infinite product:
R(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
that is,
R(x) = x * (1-x) * (1 - x^4*(1-x)) * (1 - x^16*(1-x)^4*(1 - x^4*(1-x))) * (1 - x^64*(1-x)^16*(1 - x^4*(1-x))^4*(1 - x^16*(1-x)^4*(1 - x^4*(1-x)))) * ...

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^4 + x^5 - subst(x*Ser(A), x, x^3*R), #A+3) ); H=A; A[n]}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* Using Infinite Product Formula for Series Reversion */
{F(n) = my(G=x); if(n==0, G=x, if(n==1, G=1-x, G = 1 - (1 - F(n-1))^4 * F(n-1) )); G}
{a(n) = my(A, R = prod(k=0, #binary(n), F(k) +x*O(x^n)));
A = serreverse(R); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A273162, A273203.

Sequence in context: A071735 A148363 A365267 * A364161 A148364 A172405

Adjacent sequences: A350431 A350432 A350433 * A350435 A350436 A350437

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2021

Status

approved