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A093334 Denominators of the coefficients of Euler-Ramanujan’s harmonic number expansion into negative powers of a triangular number. 2
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..inf,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.

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:0707.3950v2 [math.CA] 28 Jul 2007.

FORMULA

R_n = (-1)^(n-1)/(2*n*8^n)*(1+sum(i=1..n,(-4)^i*binom(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 A133386 A305624

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 November 11 18:50 EST 2019. Contains 329031 sequences. (Running on oeis4.)