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%I #52 Feb 09 2020 12:39:57
%S 12,120,630,1680,2310,360360,30030,1166880,17459442,193993800,
%T 223092870,486748080,579462,180970440,231415950150,493687360320,
%U 3085546002,15714504285480,62359143990,5382578744400,15465127383342,162015620206440,173062139765970,6139943741262240,77311562676150
%N Denominators of the coefficients of Euler-Ramanujan's harmonic number expansion into negative powers of a triangular number.
%C Previous name was: Coefficients in Ramanujan's Euler-MacLaurin asymptotic expansion.
%C 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
%H Stanislav Sykora, <a href="/A093334/b093334.txt">Table of n, a(n) for n = 1..296</a>
%H Chao-Ping Chen, <a href="http://dx.doi.org/10.1007/s11139-015-9670-3">On the coefficients of asymptotic expansion for the harmonic number by Ramanujan</a>, The Ramanujan Journal, (2016) 40: 279-290.
%H Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/Gamma/gammaFormulas.html">Collection of formulas for Euler's constant gamma</a> (see paragraph 2.1.1).
%H M. B. Villarino, <a href="http://arxiv.org/abs/0707.3950">Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number</a>, arXiv:0707.3950 [math.CA], 2007.
%F R_n = ((-1)^(n-1)/(2*n*8^n))*(1 + Sum_{i=1..n} (-4)^i*binomial(n,i)*B_2i(1/2));
%F a(n) = denominator(R_n), and B_2i(x) is the (2i)-th Bernoulli polynomial. - _Stanislav Sykora_, Mar 05 2014
%e R_9 = 140051/17459442 = A238813(9)/a(9).
%t 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 *)
%o (PARI) Rn(nmax)= {local(n,k,v,R);v=vector(nmax);x=1/2;
%o for(n=1,nmax,R=1;for(k=1,n,R+=(-4)^k*binomial(n,k)*eval(bernpol(2*k)));
%o R*=(-1)^(n-1)/(2*n*8^n);v[n]=R);(apply(x->denominator(x), v));}
%o // _Stanislav Sykora_, Mar 05 2014; improved by _Michel Marcus_, Aug 30 2018
%Y Cf. A000217 (triangular numbers), A001620 (gamma), A238813 (numerators).
%K nonn,frac
%O 1,1
%A Kent Wigstrom (jijiw(AT)speedsurf.pacific.net.ph), Apr 25 2004
%E Title changed, terms a(5) onward added by _Stanislav Sykora_, Mar 05 2014
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