OFFSET
0,2
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.
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.
EXAMPLE
The sequence Rn+ begins 1, 1/3, 1/30, -1/105, 1/210, -1/231, 191/30030, -29/2145, 2833/72930, ...
PROG
(PARI) a(n) = my(p=binomial(x+2, 2)^n); denominator(sum(k=0, poldegree(p), bernfrac(k)*polcoef(p, k, x)));
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Michel Marcus, May 25 2019
STATUS
approved