bibkey: hanna2003a088713 authors: Paul D. Hanna year: 2003 title: “OEIS A088713, A(x/A(x)) = 1/(1-x)” doi: null url: https://oeis.org/A088713 claim: “G.f. A(x) satisfies A(x/A(x)) = 1/(1-x).” strata_touched:

• D5/S1/Recurrence/Algebraic/AntidiagonalArraySourceSeries license: CC-BY-SA-4.0 triage: anchor

Hanna’s quotient-composition source

Paul D. Hanna’s entry is dated October 12, 2003. Its NAME reads:

G.f. A(x) satisfies A(x/A(x)) = 1/(1-x).

Its generating-function example reads:

G.f.: A(x) = 1 + x + 2x^2 + 6x^3 + 24x^4 + 118x^5 + 674*x^6 +…

The offset is 0,3, the first coefficient is one, and the keyword nonn specifies nonnegative coefficients. The formal interpretation uses a rational power series F with constant coefficient one. The quotient x/F(x) is X times the formal multiplicative inverse of F, and has zero constant coefficient, so the substitution is defined. The right-hand side is the actual unit inverse of 1-X. The theorem constructs the normalized rational solution from this equation independently of the array, proves uniqueness, and then constructs its unique natural coefficient sequence. The other functional equations, reversion formulas, and programs in the entry are not used as premises.

Verified locator

• URL: https://oeis.org/A088713
• Immutable export: https://github.com/oeis/oeisdata/blob/892a05dd4941ae76caee5906ea33ceedf3389b1d/seq/A088/A088713.seq
• Revision line: %I A088713 #19 Jan 15 2018 08:52:24.
• Author line: %A A088713 _Paul D. Hanna_, Oct 12 2003.
• License: OEIS Foundation, Creative Commons Attribution-ShareAlike 4.0, https://oeis.org/wiki/The_OEIS_End-User_License_Agreement and https://creativecommons.org/licenses/by-sa/4.0/.

The immutable export was read directly on September 21, 2026 (Singapore). This establishes the fixed source equation and normalization. The array identification is Kurkov’s separate conjecture, quoted in kurkov2025a392095.md; the defining equation alone does not assert it.