OEIS A000366: Genocchi numbers of the second kind and the unexpected integer

Explore A000366, the integers you get by dividing the Genocchi numbers of the second kind by 2^(n-1). Despite the division, every term is a positive integer, a mystery that has driven a century of study starting with Delac and Marcel in 1901. We trace two complementary viewpoints: a concrete Delac grid counting problem (2n rows, n columns, two cells per column and one per row) and Fagin’s algebraic picture in terms of nested subsets, linked through Euler characteristics of degenerate flag varieties of type A. We’ll see striking arithmetic structure: a_n ≡ 3 mod 4 if n even (n>1), a_n ≡ 2 mod 4 if n odd (n>1), and a_n modulo 36 alternates 2,7 for n>2. A deep formula ties a_n to Bernoulli numbers, and the generating function unfolds as a rich continued fraction with nested products—a true bridge between combinatorics, topology, and number theory.

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