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A238813 Numerators of the coefficients of Euler-Ramanujan’s harmonic number expansion into negative powers of a triangular number. 2
1, -1, 1, -1, 1, -191, 29, -2833, 140051, -6525613, 38899057, -532493977, 4732769, -12945933911, 168070910246641, -4176262284636781, 345687837634435, -26305470121572878741, 1747464708706073081, -2811598717039332137041, 166748874686794522517053 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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 numerators of R_n (denominators are listed in A093334).

REFERENCES

Chao-Ping Chen, On the coefficients of asymptotic expansion for the harmonic number by Ramanujan, The Ramanujan Journal, 2015; DOI 10.1007/s11139-015-9670-3

LINKS

Stanislav Sykora, Table of n, a(n) for n = 1..296

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.

EXAMPLE

R_9 = 140051/17459442 = a(9)/A093334(9).

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); return (v); }

// returns an array v[1..nmax] of the rational coefficients

CROSSREFS

Cf. A000217 (triangular numbers), A001620 (gamma), A093334 (denominators).

Sequence in context: A207198 A217343 A221112 * A103494 A104642 A115016

Adjacent sequences:  A238810 A238811 A238812 * A238814 A238815 A238816

KEYWORD

sign,frac

AUTHOR

Stanislav Sykora, Mar 05 2014

STATUS

approved

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Last modified June 27 17:44 EDT 2017. Contains 288790 sequences.