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A189731 a(n) = numerator of B(0,n) where B(n,n) = 0, B(n-1,n) = 1/n, and B(m,n) = B(m-1,n+1) - B(m-1,n).

%I A189731 #29 Nov 06 2022 09:10:40
%S A189731 0,1,1,3,2,17,4,23,25,61,18,107,40,421,1363,1103,210,5777,492,7563,
%T A189731 24475,19801,2786,103681,33552,135721,146401,355323,39650,1860497,
%U A189731 97108,2435423,2627065,6376021,20633238,11128427,1459960,43701901
%N A189731 a(n) = numerator of B(0,n) where B(n,n) = 0, B(n-1,n) = 1/n, and B(m,n) = B(m-1,n+1) - B(m-1,n).
%C A189731 Square array B(m,n) begins:
%C A189731     0,    1/1,   1/1,   3/2,   2/1,  17/6,  ...
%C A189731    1/1,    0,    1/2,   1/2,   5/6,   7/6,  ...
%C A189731   -1/1,   1/2,    0,    1/3,   1/3,   7/12, ...
%C A189731    3/2,  -1/2,   1/3,    0,    1/4,   1/4,  ...
%C A189731   -2/1,   5/6,  -1/3,   1/4,    0,    1/5,  ...
%C A189731   17/6,  -7/6,   7/12, -1/4,   1/5,    0,   ...
%C A189731 The inverse binomial transform of B(0,n) gives B(n,0) and thus it is an eigensequence in the sense that it remains the same (up to a sign) under inverse binomial transform.
%C A189731 The bisection of B(0,n) (odd part) gives A175385/A175386, and thus a(2*n+1) = A175385(n+1).
%H A189731 Alois P. Heinz, <a href="/A189731/b189731.txt">Table of n, a(n) for n = 0..1000</a>
%F A189731 Numerator of (A000204(n) - 1)/n. - _Artur Jasinski_, Oct 21 2022
%p A189731 B:= proc(m, n) option remember;
%p A189731       if m=n then 0
%p A189731     elif n=m+1 then 1/n
%p A189731     elif n>m then B(m, n-1) +B(m+1, n-1)
%p A189731     else B(m-1, n+1) -B(m-1, n)
%p A189731       fi
%p A189731     end:
%p A189731 a:= n-> numer(B(0, n)):
%p A189731 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 29 2011
%t A189731 Rest[Numerator[Abs[CoefficientList[Normal[Series[Log[1 - x^2/(1 + x)], {x, 0, 40}]], x]]]] (* _Vaclav Kotesovec_, Jul 07 2020 *)
%t A189731 Table[Numerator[(LucasL[n]-1)/n],{n,1,38}] (* _Artur Jasinski_, Oct 21 2022 *)
%Y A189731 Cf. A000204, A242926 (denominators).
%Y A189731 Cf. A174341, A177690, A181722.
%K A189731 nonn,easy
%O A189731 0,4
%A A189731 _Paul Curtz_, Apr 26 2011