A249276 - OEIS

%I #16 Apr 13 2018 07:45:33

%S -1,1,1,1,73,11,3625,5525,5233001,1212281,927777937,772193,

%T 43791735453787,6889178449747,158996102434867,107876982981287,

%U 782501215247703271,6225541612992329,235541803917995571502409

%N Numerators of fractions appearing in a generalization of Carleman's inequality.

%H Steven R. Finch, <a href="/A219245/a219245.pdf">Carleman's inequality</a>, 2013. [Cached copy, with permission of the author]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarlemansInequality.html">Carleman's Inequality</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Carleman&#39;s_inequality">Carleman's inequality</a>

%F sum_{k >= 1} (a_1 a_2 ... a_k)^(1/k) < e*sum_{k >= 1} (1-sum_{j=1..m} b_j/(k+1)^j)*a_k, where a_k >= 0 for all k and a_l > 0 for at least one l, m being any positive integer.

%F b(0) = -1, b(n) = (-1/n)*sum_{k=1..n} b(n-k)/(k+1).

%e Fractions begin -1, 1/2, 1/24, 1/48, 73/5760, 11/1280, 3625/580608, ...

%t b[0] = -1; b[n_] := b[n] = (-1/n)*Sum[b[n - k]/(k + 1), {k, 1, n}]; Table[b[n] // Numerator, {n, 0, 20}]

%t CoefficientList[Series[-(1 - x)^(-(1 - x)/x)/E, {x, 0, 20}], x] // Numerator (* _Eric W. Weisstein_, Apr 13 2018 *)

%Y Cf. A249277 (denominators), A219245, A219246, A219336.

%K sign,frac

%O 0,5

%A _Jean-François Alcover_, Oct 24 2014