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 A093334 Denominators of the coefficients of Euler-Ramanujan's harmonic number expansion into negative powers of a triangular number. 4
 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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000217 (triangular numbers), A001620 (gamma), A238813 (numerators). Sequence in context: A164877 A121032 A188251 * A001816 A354697 A133386 Adjacent sequences: A093331 A093332 A093333 * A093335 A093336 A093337 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

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Last modified March 1 01:46 EST 2024. Contains 370428 sequences. (Running on oeis4.)