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A093334
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Denominators of the coefficients of Euler-Ramanujan's harmonic number expansion into negative powers of a triangular number.
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4
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12, 120, 630, 1680, 2310, 360360, 30030, 1166880, 17459442, 193993800, 223092870, 486748080, 579462, 180970440, 231415950150, 493687360320, 3085546002, 15714504285480, 62359143990, 5382578744400, 15465127383342, 162015620206440, 173062139765970, 6139943741262240, 77311562676150
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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R_9 = 140051/17459442 = A238813(9)/a(9).
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MATHEMATICA
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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 *)
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PROG
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(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)); }
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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Kent Wigstrom (jijiw(AT)speedsurf.pacific.net.ph), Apr 25 2004
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EXTENSIONS
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STATUS
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approved
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