Metadata
bibkey: matharkimberling2026a027182
authors: R. J. Mathar and Clark Kimberling
year: 2026
title: "OEIS A027182: self-convolution of the four-diamondized Pascal row"
doi: null
url: https://oeis.org/A027182
claim: "The self-convolution sequence of A027170 satisfies the six-term integer recurrence in the A027182 FORMULA field for n at least 6."
strata_touched:
- D5/S1/Recurrence/MatharA027182SelfConvolution
license: citation-only
triage: anchor
Verified locator
Source: https://oeis.org/A027182
Underlying array: https://oeis.org/A027170
The source is unpaginated. The quoted entries are A027182 revision 8 and A027170 revision 32. Neither has an arXiv identifier or a DOI.
Source statement
A027170, definition:
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.
A027182, name and comment:
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
A027182, FORMULA:
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 formal proposition asserts the recurrence at every natural index at least 6, with integer coefficient arithmetic. Natural subtraction is used in the row definition and sequence indices. The polynomial expression also defines the natural extension a(0)=1. The source example prints 55+1010+5*5=120; the defining expression instead evaluates to 150.
Mathematical relation
Row symmetry identifies the central self-convolution with a square sum. Vandermonde and the binomial row sums give a combination of three central binomial coefficients, minus 722^n plus 16n+40. A four-shift combination of the central-binomial recurrence gives the six-term operator.
Computed evidence
The exact checker is in the-omega-institute/trureturing-experiments,
commit cf3aaa31442cac5cd2997b9c5577b98e339a310d, directory
docs/reports/mathar-a027182-recurrence/. The check.py SHA-256 is
ed87e906c89a49cda14ed1a5cfbd18654a26f99e85b08de1ecccf71d48b7c0c1.
Running python3 check.py exits 0 with ALL_CHECKS_OK: it checks the
closed form for n=0..25, the recurrence for n=5..30, and an exact symbolic
operator identity. These finite checks do not prove the universal claim.
Literature boundary
Issue #23023 records the source and literature screen. No published settlement was found in that screen. Completeness and continued external open status are ASSUMED-UNVERIFIED. Three identified Mathar works could not be read in full; several search services returned rate limits or access restrictions. Minimal annihilator order, parameterized diamondization families and a formal D-finiteness API remain open here.