OFFSET
1,1
LINKS
Philippe Flajolet and Linas Vepstas, On Differences of Zeta Values, arXiv:math/0611332 [math.CA] 2007
FORMULA
zeta(s)-1/(s-1) = Sum_{n>=0} (-1)^n*b(n)*binomial(s,n).
b(n) = n*(1-EulerGamma - H(n-1)) - 1/2 + Sum_{k=2..n} binomial(n,k)*(-1)^k*zeta(k), where H(n) is the n-th harmonic number.
EXAMPLE
b(1) = 1/2 - EulerGamma < 0,
b(2) = -1/2 - 2*EulerGamma + Pi^2/6 < 0,
b(3) = -1/2 + 3*(-1/2 - EulerGamma) + Pi^2/2 - zeta(3) > 0, so a(1) = 3.
MATHEMATICA
nmax = 2000; $MaxExtraPrecision = 1000; b[n_] := b[n] = n*(1 - EulerGamma - HarmonicNumber[n-1]) - 1/2 + Sum[Binomial[n, k]*(-1)^k*Zeta[k], {k, 2, n}]; Reap[ For[n = 1, n <= nmax, n++, If[b[n] < 0 < b[n+1] || b[n] > 0 > b[n+1], Print[n+1]; Sow[n+1]]]][[2, 1]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Jean-François Alcover, Apr 03 2015
STATUS
approved