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2^floor(log[2](n+1))*(2*n+1)!*Bernoulli(2*n,1/2).
3

%I #9 Aug 11 2014 22:46:10

%S 1,-1,7,-465,48006,-12072375,6301495035,-12203470904625,

%T 20180112406353900,-53495387545025175750,216267236072968468547250,

%U -1280630367874799320798794375,10743714652441927865738713818750,-124178158916511109662405449217796875,1930915681227482441797773554892002071875

%N 2^floor(log[2](n+1))*(2*n+1)!*Bernoulli(2*n,1/2).

%C Motivated by the original definition of A033473 which turned out to produce a non-integer sequence.

%H M. F. Hasler, <a href="/A238164/b238164.txt">Table of n, a(n) for n = 0..50</a>

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%o (PARI) n->(2*n+1)!*subst(bernpol(2*n,x),x,1/2)<<(log(n+1)\log(2))

%K sign

%O 0,3

%A _M. F. Hasler_, Feb 18 2014