A238813 Numerators of the coefficients of Euler-Ramanujan’s harmonic number expansion into negative powers of a triangular number.

1, -1, 1, -1, 1, -191, 29, -2833, 140051, -6525613, 38899057, -532493977, 4732769, -12945933911, 168070910246641, -4176262284636781, 345687837634435, -26305470121572878741, 1747464708706073081, -2811598717039332137041, 166748874686794522517053

Offset: 1

Author

Stanislav Sykora, Mar 05 2014

Keywords

• sign
• frac

Comments

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 numerators of R_n (denominators are listed in A093334).

Examples

			R_9 = 140051/17459442 = a(9)/A093334(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.
• Feng, L. and Wang, W., Riordan Array Approach to the Coefficients of Ramanujan's Harmonic Number Expansion, Results Math (2017) 71: 1413.
• 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 (also arXiv:0707.3950v2 [math.CA] 28 Jul 2007).

Crossrefs

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

Cf. A212196.

Programs

Maple

a := n -> - numer(add(binomial(n,k)*bernoulli(n+k), k=0..n)/2^n);
seq(a(n), n=1..21); # Peter Luschny, Aug 13 2017

Mathematica

Table[Numerator[-Sum[Binomial[n,k]*BernoulliB[n+k]/2^n,{k,0,n}]], {n,1,25}] (* 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);return (v);}
// returns an array v[1..nmax] of the rational coefficients

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.

From Peter Luschny, Aug 13 2017: (Start)

a(n) = -numerator(A212196(n)/2^n), A212196 the Bernoulli median numbers.

a(n) = -numerator((Sum_{k=0..n} binomial(n,k)*bernoulli(n+k))/2^n).

a(n) = -numerator(I(n)/2^n) with I(n) = (-1)^n*Integral_{x=0..1} S(n,x)^2 and S(n,x) = Sum_{k=0..n} Stirling2(n,k)*k!*(-x)^k. (End)