A093334
Denominators of the coefficients of Euler-Ramanujan's harmonic number expansion into negative powers of a triangular number.
Original entry on oeis.org
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
Offset: 1
Kent Wigstrom (jijiw(AT)speedsurf.pacific.net.ph), Apr 25 2004
R_9 = 140051/17459442 = A238813(9)/a(9).
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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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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
A331777
Numerators of coefficients in asymptotic expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k, where H_k are the harmonic numbers A001008/A002805 and gamma is the Euler-Mascheroni constant A001620.
Original entry on oeis.org
1, 1, 1, 0, -1, 1, -1, -43, 1831, 949, -137309, -85511, 3404045159, 777985057, -21024051077, -2192231411, 467347169033357, 10187765700589, -11741590582705819219, -3086703970985605357, 169597995722575162268081, 19606186988235984155519, -62715098968866173387571821
Offset: 0
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Numerator[CoefficientList[Series[Exp[2*(HarmonicNumber[k] - EulerGamma)]/k^2, {k, Infinity, 25}], 1/k]] (* Vaclav Kotesovec, Feb 10 2020 *)
A331778
Denominators of coefficients in asymptotic expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k, where H_k are the harmonic numbers A001008/A002805 and gamma is the Euler-Mascheroni constant A001620.
Original entry on oeis.org
1, 1, 3, 1, 90, 90, 567, 5670, 340200, 113400, 11226600, 5613300, 91945854000, 18389170800, 137918781000, 13135122000, 562708626480000, 11483849520000, 2020686677689680000, 505171669422420000, 3334133018187972000000, 370459224243108000000, 115027589127485034000000
Offset: 0
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Denominator[CoefficientList[Series[Exp[2*(HarmonicNumber[k] - EulerGamma)]/k^2, {k, Infinity, 25}], 1/k]] (* Vaclav Kotesovec, Feb 10 2020 *)
A308402
Denominators of the sequence of rational numbers Rn+ related to Bernoulli numbers.
Original entry on oeis.org
1, 3, 30, 105, 210, 231, 30030, 2145, 72930, 969969, 9699690, 10140585, 20281170, 22287, 6463230, 7713865005, 15427730010, 90751353, 436514007930, 1641030105, 134564468610, 368217318651, 3682173186510, 3762220429695, 127915494609630, 1546231253523, 819502564367190, 54496920530418135
Offset: 0
The sequence Rn+ begins 1, 1/3, 1/30, -1/105, 1/210, -1/231, 191/30030, -29/2145, 2833/72930, ...
- Frédéric Chapoton, Ramanujan-Bernoulli numbers as moments of Racah polynomials, arXiv:1905.09012 [math.NT], 2019.
- Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.
- M. B. Villarino, Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number, arXiv:0707.3950v2 [math.CA], 28 Jul 2007.
- 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.
Cf.
A238813 (numerators of Rn+, for n >0, up to sign).
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a(n) = my(p=binomial(x+2, 2)^n); denominator(sum(k=0, poldegree(p), bernfrac(k)*polcoef(p, k, x)));
Showing 1-4 of 4 results.
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