A308402 Denominators of the sequence of rational numbers Rn+ related to Bernoulli numbers.

1, 3, 30, 105, 210, 231, 30030, 2145, 72930, 969969, 9699690, 10140585, 20281170, 22287, 6463230, 7713865005, 15427730010, 90751353, 436514007930, 1641030105, 134564468610, 368217318651, 3682173186510, 3762220429695, 127915494609630, 1546231253523, 819502564367190, 54496920530418135

Offset: 0

Author

Michel Marcus, May 25 2019

Keywords

• nonn
• frac

Comments

The sequence Rn+ is defined by Rn+ = psi(binomial(x+2, 2)^n) where the linear form psi is defined by psi(x^n) = Bernoulli(n).

The companion sequence Rn- is defined by Rn+ = psi(binomial(x+1, 2)^n), and differs at n=1 with value -1/6 instead of 1/3.

Examples

			The sequence Rn+ begins 1, 1/3, 1/30, -1/105, 1/210, -1/231, 191/30030, -29/2145, 2833/72930, ...
		

Links

• Frédéric Chapoton, Ramanujan-Bernoulli numbers as moments of Racah polynomials, arXiv:1905.09012 [math.NT], 2019.
• Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.
• M. B. Villarino, Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number, arXiv:0707.3950v2 [math.CA], 28 Jul 2007.
• M. B. Villarino, Ramanujan’s Harmonic Number Expansion into Negative Powers of a Triangular Number, Journal of Inequalities in Pure and Applied Mathematics, Volume 9, Issue 3, Article 89.

Crossrefs

Cf. A238813 (numerators of Rn+, for n >0, up to sign).

Programs

PARI

a(n) = my(p=binomial(x+2, 2)^n); denominator(sum(k=0, poldegree(p), bernfrac(k)*polcoef(p, k, x)));