A255932 -id:A255932 - cp's OEIS frontend

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

2, -2, 26, -82, 1342, -100886, 1195742, -57242642, 31945440878, -276741323122, 26497552755742, -9169807783193206, 418093081574417342, -66910282127782482482, 37158050152167281792026, -2626016090388858294953362, 632184834985453539204543742, -1543534415494449734887808117378

Offset: 1

Peter Luschny, Mar 30 2024

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.

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.

			r(n) = 2/9, -2/81, 26/2187, -82/6561, 1342/59049, -100886/1594323, ...

Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. 39 (1938), 350-360.
Štefan Porubský, Further Congruences Involving Bernoulli Numbers, Journal of Number Theory 16, 87-94 (1983).
Georgy Feodosevich Voronyi, On Bernoulli numbers, Comm. Charkou Math. Sot. 2, 129-148 (1890) (in Russian).

Digital Library of Mathematical Functions, Voronoi's congruence.

Cf. A371640 (denominator), A371638.

Cf. A002425, A255932, A292608.

Voronoi := (a, k) -> ((a^k - 1) * bernoulli(k)) / (k * a^(k - 1)):
VoronoiList := (a, len) -> local k; [seq(Voronoi(a, 2*k), k = 1..len)]:
numer(VoronoiList(3, 18));

a(n) = Voronoi(3, 2*n) * 3^(2*n + valuation(n, 3)).

A255931 a(n) is the numerator of Gamma(n+1/2)^2/(2nPi), the value of an integral with sinh in the denominator.

Original entry on oeis.org

1, 9, 75, 11025, 178605, 36018675, 2608781175, 4108830350625, 131939107925625, 85734032330071125, 17185776480709711875, 33334677780416604466875, 4807886218329317951953125, 6509191098729563747237109375

Offset: 1

Jean-François Alcover, Mar 11 2015

			1/8, 9/64, 75/128, 11025/2048, 178605/2048, 36018675/16384, 2608781175/32768, ...

MathOverflow, An identity involving an infinite integral with a sinh in the denominator

Cf. A255932 (denominators).

a[n_] := Gamma[n+1/2]^2/(2*n*Pi) // Numerator; Array[a, 15]
Table[(2*n)!^2 / (n * 2^(4*n+1) * n!^2), {n, 1, 20}] // Numerator (* Vaclav Kotesovec, Mar 11 2015 *)

Integral_{-infinity..infinity} (prod_{j=1..n-1} j^2+x^2)*x/sinh(2*Pi*x) dx = Gamma(n+1/2)^2/(2*n*Pi).

The n-th fraction also equals the n-th coefficient in the expansion of 2F1(1/2,1/2; 1; x) * n!*(n-1)!/2.

cp's OEIS Frontend

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

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A255931 a(n) is the numerator of Gamma(n+1/2)^2/(2nPi), the value of an integral with sinh in the denominator.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Formula

A255931 a(n) is the numerator of Gamma(n+1/2)^2/(2*n*Pi), the value of an integral with sinh in the denominator.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

A255931 a(n) is the numerator of Gamma(n+1/2)^2/(2nPi), the value of an integral with sinh in the denominator.