Square Exponents and the Parity of OEIS A397902

Abstract

The zero-constant integer series of OEIS A397902 exists uniquely and has the conjectured odd support at every index above two.

The source defines A in the ring of integer formal power series, with constant coefficient zero, by [X^(m-1)](1-A)^(m squared)/(1-m squared X)=0 for every natural m>1. Here invOfUnit means the unique multiplicative inverse of the denominator with constant coefficient one; it does not require passage to rational coefficients. The sequence term a(n) is [X^n]A.

Definition 1.1 (Construction over the integers).

$$\mathrm{generatingSeries}\left(\right)=1-\mathrm{solution}\left(\right)$$

Formalization. D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.generatingSeries (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A397902: square-exponent coefficient equations and dyadic parity support. URL: https://oeis.org/A397902.

Commentary.

Put F=1-A. For n>0 set e=(n+1) squared and g=(1-eX) inverse. Let R be [X^n](F^e g), and T be [X^(n-1)](F^(e-1) F’ g + F^e g squared). Differentiation gives nR=eT. Since e-n(n+2)=1, the residual N=R-(n+2)T satisfies R=eN. Changing only the nth coefficient of F changes N by exactly that coefficient difference. Iterating the correction F_n -> F_n-N stabilizes each coefficient and defines F over the integers, with constant coefficient one. The displayed generatingSeries is A=1-F.

Definition 1.2 (The exact source equation).

$$\forall A:\mathrm{PowerSeries}\left(\mathbb{Z}\right),\mathrm{DefiningEquation}\left(A\right)⟺(\mathrm{constantCoeff}\left(A\right)=0\wedge (\forall m:\mathbb{N},1<m⟹\mathrm{coeff}\left(m-1,(1-A{)}^{m^2}\cdot \mathrm{invOfUnit}\left(1-m^2\cdot X,1\right)\right)=0))$$

Formalization. D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.DefiningEquation (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A397902: square-exponent coefficient equations and dyadic parity support. URL: https://oeis.org/A397902.

Commentary.

The predicate includes the zero constant term, integer coefficients, the square exponent, and every m>1. Multiplication by invOfUnit expresses precisely the division in the OEIS NAME.

Theorem 1.3 (The constructed series satisfies every row).

$$\mathrm{DefiningEquation}\left(\mathrm{generatingSeries}\left(\right)\right)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.generating_equation (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A397902: square-exponent coefficient equations and dyadic parity support. URL: https://oeis.org/A397902.

Commentary.

Coefficient stabilization makes every positive normalized residual zero. The identity R=eN then makes every original row zero. The geometric series used in the construction equals the denominator’s unit inverse.

Theorem 1.4 (Integer existence and uniqueness).

$$\exists !A:\mathrm{PowerSeries}\left(\mathbb{Z}\right),\mathrm{DefiningEquation}\left(A\right)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.integer_exists_unique (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A397902: square-exponent coefficient equations and dyadic parity support. URL: https://oeis.org/A397902.

Commentary.

The preceding construction supplies an integer witness. For uniqueness, equality of lower coefficients and the unit multiplier of the normalized residual force the next coefficients to agree. Induction proves equality in all degrees. No integrality assumption about a rational recurrence is used.

Definition 1.5 (Coefficient indexing).

$$\forall n:\mathbb{N},a\left(n\right)=\mathrm{coeff}\left(n,\mathrm{generatingSeries}\left(\right)\right)$$

Formalization. D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.a (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A397902: square-exponent coefficient equations and dyadic parity support. URL: https://oeis.org/A397902.

Commentary.

This is the source’s coefficient indexing, extended by a(0)=0.

Theorem 1.6 (The complete odd-support classification).

$$\forall A:\mathrm{PowerSeries}\left(\mathbb{Z}\right),\mathrm{DefiningEquation}\left(A\right)⟹\forall n:\mathbb{N},2<n⟹(\mathrm{Odd}\left(\mathrm{coeff}\left(n,A\right)\right)⟺(\exists k:\mathbb{N},1<k\wedge (n=2^k\vee n=2^k-1\vee n=2^k-2\vee n=2^k-3)))$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.hanna_conjecture (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2026). OEIS A397902: square-exponent coefficient equations and dyadic parity support. URL: https://oeis.org/A397902.

Commentary.

Reduce the exact normalized residual to ZMod 2. For the Catalan unit U, U=1+X U squared. Set H=U squared, O=(1+X)H, E=1+X+XO, and B=E squared+X O squared. Then E+XO=1+X and EO=B. These identities make the odd and even normalized rows of B vanish; repeated halving proves the needed diagonal power coefficients vanish. Normalized uniqueness identifies B with the reduction of F. For n>2, its nth coefficient equals the coefficient of U at floor((n-1)/4). The existing binary_catalan theorem identifies this coefficient as one exactly when floor((n-1)/4)+1 is a power of two. This gives precisely the four indices 2^k, 2^k-1, 2^k-2, 2^k-3 with k>1. Integer uniqueness transfers the conclusion to every series satisfying the original defining equation.

References

• Truth anchor: D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.DefiningEquation
• Truth anchor: D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.a
• Truth anchor: D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.generatingSeries
• Truth anchor: D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.generating_equation
• Truth anchor: D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.hanna_conjecture
• Truth anchor: D5/S1/Recurrence/SquareRows/SquareExponentDyadicSupport.integer_exists_unique
• Dependency: D5/S1/Recurrence/Invariants/CatalanCompositionSquareParity