Reciprocal Square-Exponent Diagonals and Catalan Parity

Abstract

The coefficients of OEIS A397356 are odd exactly when their index plus one is a power of two.

OEIS A397356 has two conjectures. Only the parity conjecture is proved here; the second conjecture, on divisibility by three, remains open. The defining equation and both conjectures are quoted in hanna2026a397356.

Write R for reciprocalSeries and G for generatingSeries, both power series over the integers, and a(n)=coeff(n,G). The coefficient function r defines R=PowerSeries.mk(r), and G=PowerSeries.invOfUnit(R,1). Write v for one plus the reduction modulo two of D5.S1.Recurrence.Invariants.CatalanCompositionSquareParity.catalanSeries. This series over ZMod(2) satisfies v^2=v+X, where X is the indeterminate. Write S for R.map(Int.castRingHom(ZMod(2))). The notation coeff(n,F) extracts the coefficient of X^n in F. All indices and exponents are natural numbers, and subtraction in an exponent is truncated natural subtraction.

Theorem 1.1 (The binary Catalan diagonal).

$$\forall n:\mathbb{N},(1<n)⟹(\mathrm{coeff}\left(n,(v{)}^{(n{)}^2}\right)=\mathrm{coeff}\left(n,(v{)}^{(n{)}^2-1}\right))$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.v_diagonal (✓ std3). ∎

Source. Repository-derived.

Commentary.

The powers v^(n^2) and v^(n^2-1) have equal coefficients at n for n>1. The residual diagonal vanishes after splitting on the parity of n. A coefficient at an odd degree of a square vanishes in the even case; square extraction in the odd case reduces the residual to a Frobenius diagonal.

Theorem 1.2 (The defining inverse pair).

$$(R\ast G=1)\wedge ((a\left(0\right)=1)\wedge ((a\left(1\right)=1)\wedge (\forall n:\mathbb{N},(1<n)⟹(\mathrm{coeff}\left(n,(R{)}^{(n{)}^2}\right)=\mathrm{coeff}\left(n,(R{)}^{(n{)}^2-1}\right)))))$$

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

Source. Repository-derived.

Commentary.

R and G are inverse, and a(0)=a(1)=1. The strict-prefix recurrence for r cancels the difference of the square-exponent diagonals of R for n>1. Since R is the reciprocal of G, this is the entry’s defining relation and makes a its coefficient sequence.

Theorem 1.3 (Uniqueness of the inverse pair).

$$\forall B:\mathrm{PowerSeries}\left(\mathbb{Z}\right),\forall A:\mathrm{PowerSeries}\left(\mathbb{Z}\right),(\mathrm{coeff}\left(0,B\right)=1)⟹((\mathrm{coeff}\left(1,B\right)=-1)⟹((\forall n:\mathbb{N},(1<n)⟹(\mathrm{coeff}\left(n,(B{)}^{(n{)}^2}\right)=\mathrm{coeff}\left(n,(B{)}^{(n{)}^2-1}\right)))⟹((B\ast A=1)⟹((B=R)\wedge (A=G)))))$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.generating_unique (✓ std3). ∎

Source. Repository-derived.

Commentary.

Any integer series B with constant coefficient 1, linear coefficient -1, and the same square-exponent diagonal relation equals R. Strong induction compares coefficients using the diagonal multiplier. If B*A=1 as well, uniqueness of the inverse gives A=G.

Theorem 1.4 (The reciprocal modulo two).

$$S=v$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.mod_two_identity (✓ std3). ∎

Source. Repository-derived.

Commentary.

Reduction of R modulo two equals v. The initial coefficients and the diagonal relations agree, so uniqueness over ZMod(2) identifies them.

Theorem 1.5 (Hanna’s A397356 parity conjecture).

$$\forall n:\mathbb{N},\mathrm{Odd}\left(a\left(n\right)\right)⟺(\exists k:\mathbb{N},n+1=(2{)}^k)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.hanna_conjecture_a397356 (✓ std3). ∎

Resolves. Problems/oeis-a397356-reciprocal-square-exponent-diagonal-parity (proved) by D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.hanna_conjecture_a397356.

Citation. Paul D. Hanna (2026). OEIS A397356, reciprocal square-exponent diagonal generating series. URL: https://oeis.org/A397356.

Commentary.

The identity S=v identifies the reduction of G with the unit inverse of v. Multiplication of that inverse by X gives the binary Catalan series. Its coefficient description therefore yields Odd(a(n)) exactly when n+1 is a power of two, including n=0.

References

• Truth anchor: D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.generating_equation
• Truth anchor: D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.generating_unique
• Truth anchor: D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.hanna_conjecture_a397356
• Truth anchor: D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.mod_two_identity
• Truth anchor: D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity.v_diagonal
• Dependency: D5/S1/Recurrence/Invariants/CatalanCompositionSquareParity
• Dependency: D5/S1/Recurrence/Parity/StripThreeTernaryCatalanParity
• Dependency: D5/S1/Recurrence/Residue/DiagonalPowerRatioAllOdd
• Dependency: D5/S1/Recurrence/Residue/QuadraticPowerDiagonalFibbinaryParity