A093334 Denominators of the coefficients of Euler-Ramanujan's harmonic number expansion into negative powers of a triangular number.

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

Author

Kent Wigstrom (jijiw(AT)speedsurf.pacific.net.ph), Apr 25 2004

Keywords

• nonn
• frac

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

Examples

			R_9 = 140051/17459442 = A238813(9)/a(9).
		

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.

Crossrefs

Cf. A000217 (triangular numbers), A001620 (gamma), A238813 (numerators).

Programs

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 *)

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

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

Extensions

Title changed, terms a(5) onward added by Stanislav Sykora, Mar 05 2014