A369547 Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^4 ) / (x*(1+x)^3).

1, 1, 1, 5, 11, 19, 46, 150, 527, 1743, 5174, 14350, 38805, 103069, 270236, 714284, 1943183, 5486591, 16069552, 48586064, 150505974, 473652950, 1502838661, 4778097313, 15153189816, 47782620920, 149511391732, 463733630212, 1425468348936, 4344295289032, 13137603866264, 39464351087160

Offset

1,4

Comments

The radius of convergence r of g.f. A(x) solves r*(1+r)^4 = 1 where r = 0.32471795724474602596...

Links

Paul D. Hanna, Table of n, a(n) for n = 1..500

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:

(1) A(x) = A( x^2*(1+x)^4 ) / (x*(1+x)^3).

(2) R(x*(1+x)^3*A(x)) = x^2*(1+x)^4, where R(A(x)) = x.

(3) A(x) = x * Product_{n>=1} F(n), where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n)^4 for n >= 1.

(4) A(x)^4 = x^3*B(x) where B(x) is the g.f. of A369554.

Example

G.f.: A(x) = x + x^2 + x^3 + 5*x^4 + 11*x^5 + 19*x^6 + 46*x^7 + 150*x^8 + 527*x^9 + 1743*x^10 + 5174*x^11 + 14350*x^12 + ...
RELATED SERIES.
A(x)^4/x^3 = x + 4*x^2 + 10*x^3 + 36*x^4 + 123*x^5 + 344*x^6 + 976*x^7 + 3000*x^8 + 9505*x^9 + ... + A369554(n)*x^n + ...
Let R(x) be the series reversion of A(x),
R(x) = x - x^2 + x^3 - 5*x^4 + 15*x^5 - 33*x^6 + 108*x^7 - 417*x^8 + 1228*x^9 - 3767*x^10 + 14409*x^11 - 49801*x^12 + ...
then R(x) and g.f. A(x) satisfy:
(1) R(A(x)) = x,
(2) R(x*(1+x)^3*A(x)) = x^2*(1 + x)^4.
GENERATING METHOD.
Define F(n), a polynomial in x of order 6^(n-1), by the following recurrence:
F(1) = (1 + x),
F(2) = (1 + x^2 * (1+x)^4),
F(3) = (1 + x^4 * (1+x)^8 * F(2)^4),
F(4) = (1 + x^8 * (1+x)^16 * F(2)^8 * F(3)^4),
F(5) = (1 + x^16 * (1+x)^32 * F(2)^16 * F(3)^8 * F(4)^4),
...
F(n+1) = 1 + (F(n) - 1)^2 * F(n)^4
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...

Prog

(PARI) {a(n) = my(A=[1], F); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = polcoeff( subst(F, x, x^2*(1 + x)^4 ) - x*(1 + x)^3*F , #A+1) ); A[n]}
for(n=1, 35, print1(a(n), ", "))

Crossrefs

Cf. A369554, A369545, A369546, A369548, A369549.

Sequence in context: A371668 A337492 A236584 * A072743 A045452 A152085

Adjacent sequences: A369544 A369545 A369546 * A369548 A369549 A369550

Keyword

nonn

Author

Paul D. Hanna, Jan 25 2024

Status

approved