[seqfan] Conjecture Involving Doubly Infinite Series

[Paul Hanna]

Seqfans,
      Would someone like to try to show that the conjecture stated below
holds true?
It involves doubly infinite series, for which other pertinent formulas are
provided in the OEIS examples that follow.  (These sums are to be regarded
as formal power series, without addressing conditions for convergence.)

This result is surprising to me and does not appear to be a trivial
deduction from known series identities.
Thanks,
      Paul

CONJECTURE.

For fixed integer m >= 1, if A(x) satisfies:
1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(m*n + m-1))^(n-1)
then
A(x) = Sum_{n=-oo..+oo}  x^((m+1)*n + m-1) * (A(x) - x^(m*n + m-1))^(n-1).

EXAMPLES.

m = 1: A357227 ------------------------------------------------

1, 1, 5, 27, 156, 961, 6145, 40546, 273784, 1883468, ...

Given g.f. A(x), F(x) = 2*A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (F(x) - x^n)^(n-1).
(2) F(x) = Sum_{n=-oo..+oo} x^(2*n) * (F(x) - x^n)^(n-1).

m = 2: A358961 ------------------------------------------------

1, 3, 7, 33, 163, 858, 4708, 26662, 154699, 914885, ...

G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(2*n+1))^(n-1).
(2) A(x) = Sum_{n=-oo..+oo} x^(3*n+1) * (A(x) - x^(2*n+1))^(n-1).

m = 3: A358962 ------------------------------------------------

1, 2, 8, 30, 146, 748, 4002, 22114, 125220, 722850, ...

G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(3*n+2))^(n-1).
(2) A(x) = Sum_{n=-oo..+oo} x^(4*n+2) * (A(x) - x^(3*n+2))^(n-1).

m = 42: A358963 -----------------------------------------------

1, 2, 7, 31, 143, 731, 3896, 21444, 120967, 695699, ...

G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(4*n+3))^(n-1).
(2) A(x) = Sum_{n=-oo..+oo} x^(5*n+3) * (A(x) - x^(4*n+3))^(n-1).

m = 5: A358964 ------------------------------------------------

1, 2, 7, 30, 144, 728, 3879, 21338, 120301, 691482, ...

G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(5*n+4))^(n-1).
(2) A(x) = Sum_{n=-oo..+oo} x^(6*n+4) * (A(x) - x^(5*n+4))^(n-1).

m = 6: A358965 ------------------------------------------------

1, 2, 7, 30, 143, 729, 3876, 21321, 120195, 690816, ...

G.f. A(x) satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (A(x) - x^(6*n+5))^(n-1).
(2) A(x) = Sum_{n=-oo..+oo} x^(7*n+5) * (A(x) - x^(6*n+5))^(n-1).

================= END ==================