A249276 Numerators of fractions appearing in a generalization of Carleman's inequality.

-1, 1, 1, 1, 73, 11, 3625, 5525, 5233001, 1212281, 927777937, 772193, 43791735453787, 6889178449747, 158996102434867, 107876982981287, 782501215247703271, 6225541612992329, 235541803917995571502409

Offset

0,5

Links

Table of n, a(n) for n=0..18.

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]

Eric Weisstein's World of Mathematics, Carleman's Inequality

Wikipedia, Carleman's inequality

Formula

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.

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

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A113889 A099191 A051325 * A102050 A057446 A033393

Adjacent sequences: A249273 A249274 A249275 * A249277 A249278 A249279

Keyword

sign,frac

Author

Jean-François Alcover, Oct 24 2014

Status

approved