A331474 Signed Hankel Bridge

Abstract

The literal A331473 moments and their A331474 Hankel determinants are connected to a signed period-two continuant, yielding an exact all-order scalar bridge.

All indices below are natural numbers. Integer-valued sequences are written with subscripts. The symbol X is the polynomial indeterminate, coeff selects a polynomial coefficient, and det is the determinant of the displayed finite matrix. These definitions use the literal offset-zero OEIS source, not a recurrence-defined replacement for its Hankel transform.

Definition 1.1 (Unsigned Catalan-derivative moments).

$$\forall n\in \mathbb{N},b_n=\mathrm{binom}\left(2\cdot n+2,n\right)$$

Formalization. D5/S3/Constants/Moments/A331474HankelBridge.b (✓ std3).

Citation. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Commentary.

The moment b(n) is exactly binomial(2n+2,n), equivalently (n+1) times the Catalan number of index n+1. It is the coefficient sequence obtained by differentiating the Catalan power series.

Definition 1.2 (Literal alternating source sequence).

$$\forall n\in \mathbb{N},s_n=\sum _{k=0}^n(-1{)}^{n-k}\cdot \mathrm{binom}\left(2\cdot k+2,k\right)$$

Formalization. D5/S3/Constants/Moments/A331474HankelBridge.s (✓ std3).

Citation. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Commentary.

This is the full alternating binomial transform printed in A331473, with k ranging from zero through n and with no recurrence substituted for it.

Definition 1.3 (Literal Hankel determinant).

$$\forall n\in \mathbb{N},H_n=\det \left((s_{i+j}{)}_{0\le i,j\le n}\right)$$

Formalization. D5/S3/Constants/Moments/A331474HankelBridge.H (✓ std3).

Citation. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Commentary.

H(n) is the determinant of the actual (n+1)-square Hankel matrix of s. Consequently the final generating function is about PowerSeries.mk H, rather than a sequence introduced later by its recurrence.

Definition 1.4 (Period-two diagonal coefficient).

$$\forall n\in \mathbb{N},a_n=\mathrm{ite}\left(\mathrm{Even}\left(n\right),4,0\right)$$

Formalization. D5/S3/Constants/Moments/A331474HankelBridge.a (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Acknowledgement. Radica Bojičić; M. D. Petković; Paul Barry (2025). Hankel transform of linear combination of three consecutive Catalan numbers. DOI: 10.15672/hujms.1564485.

Commentary.

The source-specific monic continuant uses diagonal coefficient 4 at even indices and 0 at odd indices. This period-two tail is derived from the Catalan derivative equation.

Definition 1.5 (Signed monic continuants).

$$\begin{array}{r}{P}_0=1\\ P_1=X-4\\ \forall m\in \mathbb{N},P_{m+2}=(X-a_{m+1})\cdot P_{m+1}+P_m\end{array}$$

Formalization. D5/S3/Constants/Moments/A331474HankelBridge.P (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Acknowledgement. Radica Bojičić; M. D. Petković; Paul Barry (2025). Hankel transform of linear combination of three consecutive Catalan numbers. DOI: 10.15672/hujms.1564485.

Commentary.

The polynomials are monic: P(0)=1, P(1)=X-4, and P(m+2)=(X-a(m+1))P(m+1)+P(m). The plus sign gives signed squared norms (-1)^k; positivity is neither assumed nor true here.

Definition 1.6 (Unsigned moment functional).

$$\forall f\in \mathbb{Z}[X],B\left(f\right)=\sum _{n\in \mathbb{N}}\mathrm{coeff}\left(f,n\right)\cdot b_n$$

Formalization. D5/S3/Constants/Moments/A331474HankelBridge.B (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Acknowledgement. Radica Bojičić; M. D. Petković; Paul Barry (2025). Hankel transform of linear combination of three consecutive Catalan numbers. DOI: 10.15672/hujms.1564485.

Commentary.

B evaluates an integer polynomial by weighting every coefficient with b. The Catalan derivative equation supplies the two alternating tails used to prove orthogonality of the monic continuants.

Definition 1.7 (Literal alternating moment functional).

$$\forall f\in \mathbb{Z}[X],S\left(f\right)=\sum _{n\in \mathbb{N}}\mathrm{coeff}\left(f,n\right)\cdot s_n$$

Formalization. D5/S3/Constants/Moments/A331474HankelBridge.S (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Acknowledgement. Radica Bojičić; M. D. Petković; Paul Barry (2025). Hankel transform of linear combination of three consecutive Catalan numbers. DOI: 10.15672/hujms.1564485.

Commentary.

S evaluates a polynomial against the literal sequence s. The source relation b(n+1)=s(n+1)+s(n) transfers the unsigned Catalan identities to this signed functional.

Definition 1.8 (Continuant constant terms).

$$\forall n\in \mathbb{N},p_n=\mathrm{coeff}\left(P_n,0\right)$$

Formalization. D5/S3/Constants/Moments/A331474HankelBridge.p (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Acknowledgement. Radica Bojičić; M. D. Petković; Paul Barry (2025). Hankel transform of linear combination of three consecutive Catalan numbers. DOI: 10.15672/hujms.1564485.

Commentary.

The scalar p(n) is the constant coefficient of P(n). Its exact initial values and recurrence are included in the bridge theorem below.

Definition 1.9 (Literal functional values).

$$\forall n\in \mathbb{N},u_n=S\left(P_n\right)$$

Formalization. D5/S3/Constants/Moments/A331474HankelBridge.u (✓ std3).

Source. Repository-derived.

Acknowledgement. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Acknowledgement. Radica Bojičić; M. D. Petković; Paul Barry (2025). Hankel transform of linear combination of three consecutive Catalan numbers. DOI: 10.15672/hujms.1564485.

Commentary.

The scalar u(n) is S(P(n)). Its exact initial values and source-specific recurrence are included in the bridge theorem below.

Theorem 1.10 (Literal determinant equals the signed kernel).

$$\begin{array}{r}\forall n\in \mathbb{N},\\ H_n=\prod _{k=0}^n(-1{)}^k\cdot \sum _{k=0}^n(-1{)}^k{p}_k{u}_k\wedge \\ p_0=1\wedge p_1=-4\wedge \\ (\forall m\in \mathbb{N},p_{m+2}=-a_{m+1}\cdot p_{m+1}+p_m)\wedge \\ u_0=1\wedge u_1=-1\wedge u_2=1\wedge \\ \forall m\in \mathbb{N},2\le m\Rightarrow u_{m+1}=-(a_m+1)\cdot u_m+u_{m-1}.\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Constants/Moments/A331474HankelBridge.literal_hankel_eq_signed_kernel (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Paul Barry (2020). OEIS A331474, Hankel transform of A331473. URL: https://oeis.org/A331474.

Acknowledgement. Radica Bojičić; M. D. Petković; Paul Barry (2025). Hankel transform of linear combination of three consecutive Catalan numbers. DOI: 10.15672/hujms.1564485.

Commentary.

The Catalan derivative coefficients first give two period-two power-series tails. Their continuant error identity yields signed monomial orthogonality, which extends to polynomial orthogonality. The resulting monic coefficient matrix diagonalizes the Gram matrix with diagonal entries (-1)^k. Negative entries are intentional, so no positive-real Gram argument is used.

A division-free adjugate identity then replaces matrix inversion. For the actual adjacent-column coefficient c=-1, the Hankel determinant is converted to a Cramer determinant and then to the signed reproducing kernel. The same proof derives p(0)=1, p(1)=-4 and its recurrence, and u(0)=1, u(1)=-1, u(2)=1 and its quantified recurrence. No finite-prefix certificate or positivity hypothesis carries any part of the result.

References

• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.B
• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.H
• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.P
• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.S
• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.a
• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.b
• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.literal_hankel_eq_signed_kernel
• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.p
• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.s
• Truth anchor: D5/S3/Constants/Moments/A331474HankelBridge.u