Mathar and Kimberling’s self-convolution recurrence

Abstract

The central self-convolution of the four-diamondized Pascal row satisfies the six-term recurrence in OEIS A027182.

Definition 1.1 (The four binomial coefficients).

$$\forall n\in \mathbb{N},\forall k\in \mathbb{N},t\left(n,k\right)=\mathrm{Nat}.\mathrm{choose}\left(n+2,k+1\right)+\mathrm{Nat}.\mathrm{choose}\left(n+1,k\right)+\mathrm{Nat}.\mathrm{choose}\left(n+1,k+1\right)+\mathrm{Nat}.\mathrm{choose}\left(n,k\right)$$

Formalization. D5/S1/Recurrence/MatharA027182SelfConvolution.t (✓ std3).

Citation. R. J. Mathar and Clark Kimberling (2026). OEIS A027182: self-convolution of the four-diamondized Pascal row. URL: https://oeis.org/A027182.

Commentary.

OEIS A027170, revision 32, definition (unpaginated): “Triangular array T read by rows (4-diamondization of Pascal’s triangle). Step 1: t(n,k) = C(n+2,k+1) + C(n+1,k) + C(n+1,k+1) + C(n,k). Step 2: T(n,k) = t(n,k) - t(0,0) + 1. Domain: 0 <= k <= n, n >= 0.” The indices n and k are natural numbers, and Nat.choose is the ordinary binomial coefficient.

Definition 1.2 (The diamondized row).

$$\forall n\in \mathbb{N},\forall k\in \mathbb{N},T\left(n,k\right)=t\left(n,k\right)-t\left(0,0\right)+1$$

Formalization. D5/S1/Recurrence/MatharA027182SelfConvolution.T (✓ std3).

Citation. R. J. Mathar and Clark Kimberling (2026). OEIS A027182: self-convolution of the four-diamondized Pascal row. URL: https://oeis.org/A027182.

Commentary.

OEIS A027170, revision 32, definition (unpaginated): “Triangular array T read by rows (4-diamondization of Pascal’s triangle). Step 1: t(n,k) = C(n+2,k+1) + C(n+1,k) + C(n+1,k+1) + C(n,k). Step 2: T(n,k) = t(n,k) - t(0,0) + 1. Domain: 0 <= k <= n, n >= 0.” The subtraction is natural subtraction; the stated row domain is 0 <= k <= n. The formula defines a total natural-valued function.

Definition 1.3 (The row polynomial).

$$\forall n\in \mathbb{N},g\left(n\right)=\sum _{k\in \mathrm{Finset}.\mathrm{range}\left(n+1\right)}\mathrm{Polynomial}.C\left(T\left(n,k\right)\right)\cdot \mathrm{Polynomial}.X^k$$

Formalization. D5/S1/Recurrence/MatharA027182SelfConvolution.g (✓ std3).

Citation. R. J. Mathar and Clark Kimberling (2026). OEIS A027182: self-convolution of the four-diamondized Pascal row. URL: https://oeis.org/A027182.

Commentary.

OEIS A027182, revision 8, name and comment (unpaginated): “a(n) = self-convolution of row n of array T given by A027170.” “Let g(x) = Sum_(k=0..n) A027170(n,k)*x^k be the generating function of row n, then a(n) = [x^n] g(x)^2. - R. J. Mathar, May 06 2026” The polynomial has natural coefficients. Finset.range(n+1) includes both row endpoints 0 and n; Polynomial.C and Polynomial.X form each monomial.

Definition 1.4 (The central coefficient of the square).

$$\forall n\in \mathbb{N},a\left(n\right)=\mathrm{Polynomial}.\mathrm{coeff}\left(g{\left(n\right)}^2,n\right)$$

Formalization. D5/S1/Recurrence/MatharA027182SelfConvolution.a (✓ std3).

Citation. R. J. Mathar and Clark Kimberling (2026). OEIS A027182: self-convolution of the four-diamondized Pascal row. URL: https://oeis.org/A027182.

Commentary.

OEIS A027182, revision 8, name and comment (unpaginated): “a(n) = self-convolution of row n of array T given by A027170.” “Let g(x) = Sum_(k=0..n) A027170(n,k)*x^k be the generating function of row n, then a(n) = [x^n] g(x)^2. - R. J. Mathar, May 06 2026” Polynomial.coeff extracts the coefficient with natural index n. The source sequence starts at 1; the same expression defines a(0)=1.

Definition 1.5 (The conjectured recurrence).

$$claim\iff (\forall n\in \mathbb{N},6\le n\Rightarrow (2\cdot ((n:\mathbb{Z})+2)\cdot (a\left(n\right):\mathbb{Z})+(-17\cdot (n:\mathbb{Z})-23)\cdot (a\left(n-1\right):\mathbb{Z})+(47\cdot (n:\mathbb{Z})+36)\cdot (a\left(n-2\right):\mathbb{Z})+2\cdot (-22\cdot (n:\mathbb{Z})-17)\cdot (a\left(n-3\right):\mathbb{Z})+4\cdot (-(n:\mathbb{Z})+20)\cdot (a\left(n-4\right):\mathbb{Z})+8\cdot (2\cdot (n:\mathbb{Z})-9)\cdot (a\left(n-5\right):\mathbb{Z})+24\cdot (12\cdot (n:\mathbb{Z})-47)=0))$$

Formalization. D5/S1/Recurrence/MatharA027182SelfConvolution.claim (✓ std3).

Citation. R. J. Mathar and Clark Kimberling (2026). OEIS A027182: self-convolution of the four-diamondized Pascal row. URL: https://oeis.org/A027182.

Commentary.

OEIS A027182, revision 8, FORMULA field (unpaginated): “Conjecture: D-finite with recurrence 2*(n+2)a(n) +(-17n-23)a(n-1) +(47n+36)a(n-2) +2(-22n-17)a(n-3) +4(-n+20)a(n-4) +8(2n-9)a(n-5) +24(12*n-47)=0. - R. J. Mathar, May 06 2026” The proposition ranges over every natural n with 6 <= n. Sequence indices use natural subtraction. Each sequence value and n are cast to the integers before coefficient arithmetic. The proposition asserts the displayed recurrence; no separate D-finiteness predicate is asserted.

Theorem 1.6 (The recurrence holds).

$$\forall n\in \mathbb{N},6\le n\Rightarrow (2\cdot ((n:\mathbb{Z})+2)\cdot (a\left(n\right):\mathbb{Z})+(-17\cdot (n:\mathbb{Z})-23)\cdot (a\left(n-1\right):\mathbb{Z})+(47\cdot (n:\mathbb{Z})+36)\cdot (a\left(n-2\right):\mathbb{Z})+2\cdot (-22\cdot (n:\mathbb{Z})-17)\cdot (a\left(n-3\right):\mathbb{Z})+4\cdot (-(n:\mathbb{Z})+20)\cdot (a\left(n-4\right):\mathbb{Z})+8\cdot (2\cdot (n:\mathbb{Z})-9)\cdot (a\left(n-5\right):\mathbb{Z})+24\cdot (12\cdot (n:\mathbb{Z})-47)=0)$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/MatharA027182SelfConvolution.result (✓ std3). ∎

Resolves. Problems/mathar-a027182 (proved) by D5/S1/Recurrence/MatharA027182SelfConvolution.result.

Source. Repository-derived.

Acknowledgement. R. J. Mathar and Clark Kimberling (2026). OEIS A027182: self-convolution of the four-diamondized Pascal row. URL: https://oeis.org/A027182.

Commentary.

Symmetry of the row turns its central self-convolution into a sum of squares. Vandermonde’s identity evaluates the square and mixed binomial sums. The resulting closed form is a combination of three consecutive central binomial coefficients, an exponential and an affine term. Four consecutive central-binomial identities give an order-two recurrence for that combination. A shifted operator factorization then yields the six-term recurrence.

References

• Truth anchor: D5/S1/Recurrence/MatharA027182SelfConvolution.T
• Truth anchor: D5/S1/Recurrence/MatharA027182SelfConvolution.a
• Truth anchor: D5/S1/Recurrence/MatharA027182SelfConvolution.claim
• Truth anchor: D5/S1/Recurrence/MatharA027182SelfConvolution.g
• Truth anchor: D5/S1/Recurrence/MatharA027182SelfConvolution.result
• Truth anchor: D5/S1/Recurrence/MatharA027182SelfConvolution.t