OFFSET
1,6
COMMENTS
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 numerators of R_n (denominators are listed in A093334).
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.
Feng, L. and Wang, W., Riordan Array Approach to the Coefficients of Ramanujan's Harmonic Number Expansion, Results Math (2017) 71: 1413.
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 (also arXiv:0707.3950v2 [math.CA] 28 Jul 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.
From Peter Luschny, Aug 13 2017: (Start)
a(n) = -numerator((Sum_{k=0..n} binomial(n,k)*bernoulli(n+k))/2^n).
a(n) = -numerator(I(n)/2^n) with I(n) = (-1)^n*Integral_{x=0..1} S(n,x)^2 and S(n,x) = Sum_{k=0..n} Stirling2(n,k)*k!*(-x)^k. (End)
EXAMPLE
R_9 = 140051/17459442 = a(9)/A093334(9).
MAPLE
a := n -> - numer(add(binomial(n, k)*bernoulli(n+k), k=0..n)/2^n);
seq(a(n), n=1..21); # Peter Luschny, Aug 13 2017
MATHEMATICA
Table[Numerator[-Sum[Binomial[n, k]*BernoulliB[n+k]/2^n, {k, 0, n}]], {n, 1, 25}] (* 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); return (v); }
// returns an array v[1..nmax] of the rational coefficients
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Stanislav Sykora, Mar 05 2014
STATUS
approved