OFFSET
1,1
COMMENTS
Previous name was: Coefficients in Ramanujan's Euler-MacLaurin asymptotic expansion.
Explicitly, H_k = Sum_{i=1..k} 1/i = log(2*m)/2 + gamma + Sum_{n>=1} R_n/m^n, where m = k(k+1)/2 is the k-th triangular number. This sequence lists the denominators of R_n (numerators are listed in A238813). A few starting numerical terms were given by Euler and Ramanujan; the form of the general term and the behavior of the series were determined by Villarino. - Stanislav Sykora, Mar 05 2014
LINKS
Stanislav Sykora, Table of n, a(n) for n = 1..296
Chao-Ping Chen, On the coefficients of asymptotic expansion for the harmonic number by Ramanujan, The Ramanujan Journal, (2016) 40: 279-290.
Xavier Gourdon and Pascal Sebah, Collection of formulas for Euler's constant gamma (see paragraph 2.1.1).
M. B. Villarino, Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number, arXiv:0707.3950 [math.CA], 2007.
FORMULA
R_n = ((-1)^(n-1)/(2*n*8^n))*(1 + Sum_{i=1..n} (-4)^i*binomial(n,i)*B_2i(1/2));
a(n) = denominator(R_n), and B_2i(x) is the (2i)-th Bernoulli polynomial. - Stanislav Sykora, Mar 05 2014
EXAMPLE
R_9 = 140051/17459442 = A238813(9)/a(9).
MATHEMATICA
Table[Denominator[((-1)^(n-1)/(2*n*8^n))*(1 + Sum[(-4)^j*Binomial[n, j]* BernoulliB[2*j, 1/2], {j, 1, n}])], {n, 1, 30}] (* G. C. Greubel, Aug 30 2018 *)
PROG
(PARI) Rn(nmax)= {local(n, k, v, R); v=vector(nmax); x=1/2;
for(n=1, nmax, R=1; for(k=1, n, R+=(-4)^k*binomial(n, k)*eval(bernpol(2*k)));
R*=(-1)^(n-1)/(2*n*8^n); v[n]=R); (apply(x->denominator(x), v)); }
// Stanislav Sykora, Mar 05 2014; improved by Michel Marcus, Aug 30 2018
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Kent Wigstrom (jijiw(AT)speedsurf.pacific.net.ph), Apr 25 2004
EXTENSIONS
Title changed, terms a(5) onward added by Stanislav Sykora, Mar 05 2014
STATUS
approved