|
%I #22 Aug 30 2018 15:56:23
%S 1,-1,1,-1,1,-191,29,-2833,140051,-6525613,38899057,-532493977,
%T 4732769,-12945933911,168070910246641,-4176262284636781,
%U 345687837634435,-26305470121572878741,1747464708706073081,-2811598717039332137041,166748874686794522517053
%N Numerators of the coefficients of Euler-Ramanujan’s harmonic number expansion into negative powers of a triangular number.
%C 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).
%H Stanislav Sykora, <a href="/A238813/b238813.txt">Table of n, a(n) for n = 1..296</a>
%H Chao-Ping Chen, <a href="http://dx.doi.org/10.1007/s11139-015-9670-3">On the coefficients of asymptotic expansion for the harmonic number by Ramanujan</a>, The Ramanujan Journal, (2016) 40: 279.
%H Feng, L. and Wang, W., <a href="http://dx.doi.org/10.1007/s00025-016-0597-9">Riordan Array Approach to the Coefficients of Ramanujan's Harmonic Number Expansion</a>, Results Math (2017) 71: 1413.
%H M. B. Villarino, <a href="https://www.emis.de/journals/JIPAM/article1026.html?sid=1026">Ramanujan’s Harmonic Number Expansion into Negative Powers of a Triangular Number</a>, Journal of Inequalities in Pure and Applied Mathematics, Volume 9, Issue 3, Article 89 (also arXiv:0707.3950v2 [math.CA] 28 Jul 2007).
%F 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.
%F From _Peter Luschny_, Aug 13 2017: (Start)
%F a(n) = -numerator(A212196(n)/2^n), A212196 the Bernoulli median numbers.
%F a(n) = -numerator((Sum_{k=0..n} binomial(n,k)*bernoulli(n+k))/2^n).
%F 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)
%e R_9 = 140051/17459442 = a(9)/A093334(9).
%p a := n -> - numer(add(binomial(n,k)*bernoulli(n+k), k=0..n)/2^n);
%p seq(a(n), n=1..21); # _Peter Luschny_, Aug 13 2017
%t Table[Numerator[-Sum[Binomial[n,k]*BernoulliB[n+k]/2^n,{k,0,n}]], {n,1,25}] (* _G. C. Greubel_, Aug 30 2018 *)
%o (PARI) Rn(nmax)= {local(n,k,v,R);v=vector(nmax);x=1/2;
%o for(n=1,nmax,R=1;for(k=1,n,R+=(-4)^k*binomial(n,k)*eval(bernpol(2*k)));
%o R*=(-1)^(n-1)/(2*n*8^n);v[n]=R);return (v);}
%o // returns an array v[1..nmax] of the rational coefficients
%Y Cf. A000217 (triangular numbers), A001620 (gamma), A093334 (denominators).
%Y Cf. A212196.
%K sign,frac
%O 1,6
%A _Stanislav Sykora_, Mar 05 2014
|
