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A371639 a(n) = numerator(Voronoi(3, 2*n)) where Voronoi(c, n) = ((c^n - 1) * Bernoulli(n)) / (n * c^(n - 1)).

%I A371639 #14 Mar 31 2024 11:41:10
%S A371639 2,-2,26,-82,1342,-100886,1195742,-57242642,31945440878,-276741323122,
%T A371639 26497552755742,-9169807783193206,418093081574417342,
%U A371639 -66910282127782482482,37158050152167281792026,-2626016090388858294953362,632184834985453539204543742,-1543534415494449734887808117378
%N A371639 a(n) = numerator(Voronoi(3, 2*n)) where Voronoi(c, n) = ((c^n - 1) * Bernoulli(n)) / (n * c^(n - 1)).
%C A371639 To begin with, we observe that if c = 2, then the numerator of Voronoi(2, 2*n) is the same as the numerator of Euler(2*n - 1, 1), which is equal to (-1)^n*A002425(n). Similarly, the denominator of Voronoi(2, 2*n) is A255932(n), which is equal to 2^A292608(n). The rational sequence r(n) = a(n) / A371640(n) examines the corresponding relationships in the case c = 3.
%C A371639 The function Voronoi, which is defined in the Name, was inspired by Voronoi's congruence. This congruence states that for any even integer k >= 2 and all positive coprime integers c, n: (c^k - 1)*N(k) == k*c^(k-1)*D(k)*Sum_{m=1..n-1} m^(k-1)* floor(m*c / n) mod n, where N(k) = numerator(Bernoulli(k)), D(k) = denominator( Bernoulli(k)) and gcd(N(k), D(k)) = 1.
%D A371639 Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. 39 (1938), 350-360.
%D A371639 Štefan Porubský, Further Congruences Involving Bernoulli Numbers, Journal of Number Theory 16, 87-94 (1983).
%D A371639 Georgy Feodosevich Voronyi, On Bernoulli numbers, Comm. Charkou Math. Sot. 2, 129-148 (1890) (in Russian).
%H A371639 Digital Library of Mathematical Functions, <a href="https://dlmf.nist.gov/24.10#iii">Voronoi's congruence</a>.
%F A371639 a(n) = Voronoi(3, 2*n) * 3^(2*n + valuation(n, 3)).
%e A371639 r(n) = 2/9, -2/81, 26/2187, -82/6561, 1342/59049, -100886/1594323, ...
%p A371639 Voronoi := (a, k) -> ((a^k - 1) * bernoulli(k)) / (k * a^(k - 1)):
%p A371639 VoronoiList := (a, len) -> local k; [seq(Voronoi(a, 2*k), k = 1..len)]:
%p A371639 numer(VoronoiList(3, 18));
%Y A371639 Cf. A371640 (denominator), A371638.
%Y A371639 Cf. A002425, A255932, A292608.
%K A371639 sign,frac
%O A371639 1,1
%A A371639 _Peter Luschny_, Mar 30 2024