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
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).
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((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)
Comments