bibkey: kurkov2026a396563
authors: Mikhail Kurkov
year: 2026
title: "OEIS A396563: three-parameter Stirling-minor Vandermonde sum"
doi: null
url: https://oeis.org/A396563/internal
claim: "The first-kind Stirling minor B(n,k,m) equals the normalized nested Vandermonde-square sum for n,k >= 0 and m >= 1."
strata_touched:
  - D5/S3/Combinatorics/BinomialBases/KurkovStirlingMinorVandermondeSum
license: citation-only
triage: anchor

Kurkov’s three-parameter Stirling-minor sum

Verified locator

Source: https://oeis.org/A396563/internal

The unpaginated OEIS record, approved revision 15, contains the following sole FORMULA conjecture:

%F A396563 Conjecture: B(n, k, m) = (Sum_{j_1=0..n} Sum_{j_2=0..j_1} ... Sum_{j_m=0..j_{m-1}} (-1)^(j_1 + j_2 + ... + j_m) * (Product_{1 <= q < p <= m} (j_q - j_p + p - q)^2) * Product_{t=1..m} (n+t-1)_{j_t} * (n-j_t+t)^(n+k) / (j_t+m-t)!) / (Product_{i=0..m-1} (n+i)!) where B(n, k, m) is the determinant of the k X k matrix with (i, j)-th entry |Stirling1(n+m+i, n+j)|, where (x)_{n} is the falling factorial defined as (x)_{n} = Product_{k=0..n-1} (x-k).

The source programs use one-based matrix indices. In Lean, positions are Fin k and Fin m with one added where the source starts at one. The unsigned first-kind numbers are Nat.stirlingFirst; the falling factorial is Nat.descFactorial. The existing Chain n m is the antitone-map domain, so summing over it counts each nested tuple once. All division and Vandermonde differences are rational; indices, factorials and sign exponents are natural. The empty determinant is one. The assertion includes n=0 and k=0 and assumes m>=1. No arXiv identifier or DOI is associated with this OEIS-only source.

The proof identifies the first-kind coefficient minor and the complete-coefficient corner of inverse Toeplitz matrices. Cauchy–Binet, the chain/subset bijection and Stirling inclusion–exclusion identify the nested sum with the same corner. The determinant-one inverse-corner identity is valid for arbitrary rational inverse matrices; it does not assert positivity of the signed weights.

The literature screen in issue #22587 found no identified settlement in the screened sources. Completeness is ASSUMED-UNVERIFIED, including unindexed or differently worded subsequent results. The A132393 reciprocal-power identity has a different normalization and does not supply this three-parameter identity.

The exact finite check is in the-omega-institute/trureturing-experiments, commit 7254c4bcdf4690b87752f52b771b4d6352dfc5dd, docs/reports/oeis-a396563-stirling-minor/check.py, SHA256 6041d41d7d2231c3fef7006967f8c14131831aec3a451dd0a74e1e74addb4379. The source issue records exit 0 and ALL_CHECKS_OK for 245 parameter triples and 11,956 chain instances. These are finite checks, not a universal proof. The m=0 extension, a Schur-function interpretation and separately stated positivity and integrality corollaries remain follow-up questions.