bibkey: hanna2026a392210 authors: Paul D. Hanna year: 2026 title: “OEIS A392210, G.f.: x + sq(x/(1-x) + sq(x/(1-2x) + sq(x/(1-3x) + …))), an infinitely nested square” doi: null url: https://oeis.org/A392210 claim: “A392210 %C: It appears that a(2*n-1) is even for n > 1.” strata_touched:

• D5/S1/Recurrence/Parity/NestedSquareOddEven license: citation-only triage: anchor

OEIS A392210

Paul D. Hanna’s entry of February 2, 2026 defines the generating function as the infinitely nested square x + sq(x/(1-x) + sq(x/(1-2*x) + sq(x/(1-3*x) + ...))), where sq(x) = x^2. Its terms begin 1, 1, 4, 18, 96, 579, 3834, 27404, 209052, 1688389, 14350934, 127790843, ….

Verified locator

• URL: https://oeis.org/A392210
• Locator: NAME, “G.f.: x + sq(x/(1-x) + sq(x/(1-2x) + sq(x/(1-3x) + sq(x/(1-4x) + …)))) where sq(x) = x^2, an infinitely nested square.”; COMMENTS, “It appears that a(2n-1) is even for n > 1.”; FORMULA (1), “A(x) = x + A(x/(1-x))^2.”; FORMULA, “a(n) = Sum_{k=1..n-1} b(n-k) * b(k) for n > 1 with a(1) = 1 where b(n) = Sum_{k=1..n} binomial(n-1,k-1) * a(k).”
• Sequence author Paul D. Hanna, Feb 02 2026; revision 15, Feb 04 2026. The comment carries no answer.

Reading of the statement

Substituting x/(1-x) into x/(1-k*x) gives x/(1-(k+1)*x), so iterating formula (1) unrolls into the nested square of the name. The statement to settle is that every solution of A(x) = x + A(x/(1-x))^2 has even coefficients at the odd indices 2n - 1 with n > 1.

Scope of the recorded answer

The comment holds. For any integer power series F, the coefficient of x^(2m+1) in F^2 is Σ_{i+j=2m+1} f_i f_j; swapping i and j pairs the terms, and no term is paired with itself because 2m+1 is odd, so the sum is twice an integer. Applying this with F = A(x/(1-x)) and noting that x contributes only at index one gives the claim. The functional equation does have a solution: the entry’s coefficient recurrence defines one, since substituting x/(1-x) sends the coefficient sequence to its binomial transform b.

Bounded prior-resolution evidence

The entry was read in full and records no proof and no reference to one. Its cross-references A090363, A119685, A000108 and A393172 were read and state no parity result for this sequence. Searching the catalogue by the first eight terms returns only this entry. No paper or note giving the result was found on MathDB, and google-deepmind/formal-conjectures has no file for the entry. Citation indices were not exhaustively reachable, so this is a bounded negative finding and no worldwide priority claim is made.