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%I #9 Mar 14 2014 11:43:10
%S 1,1,1,-2,1,-3,1,-4,2,1,-5,5,1,-6,9,-10,1,-7,14,-35,1,-8,20,-80,26,1,
%T -9,27,-150,117,1,-10,35,-250,325,-454,1,-11,44,-385,715,-2497,1,-12,
%U 54,-560,1365,-8172,5914,1,-13
%N Unreduced numerators in triangle that leads to the Euler numbers A198631(n)/A006519(n+1).
%C We use the array ASPEC mentioned in A191302:
%C 2, 1, 1, 1, 1, 1, 1, 1,...
%C 2, 3, 4, 5, 6, 7, 8, 9,...
%C 2, 5, 9, 14, 20, 27, 35, 44,...
%C 2, 7, 16, 30, 50, 77, 112, 156,...
%C with the first upper diagonal of the difference table of the autosequence A198631(n)/A006519(n+1), i.e., 1/2, -1/4, 1/4, -5/8, 13/4, -227/8, 2957/8,...
%C written by columns:
%C 1/2
%C 1/2,
%C 1/2, -1/4,
%C 1/2, -1/4,
%C 1/2, -1/4, 1/4,
%C 1/2, -1/4, 1/4,
%C 1/2, -1/4, 1/4, -5/8,
%C 1/2, -1/4, 1/4, -5/8,
%C etc.
%C Hence, by multiplication of this double triangle by ASPEC, the beginning of the double triangle ESPEC is obtained:
%C E(0) = 1 = 1
%C E(1) = 1/2 = 1/2
%C E(2) = 0 = 1/2 -2/4
%C E(3) = -1/4 = 1/2 -3/4
%C E(4) = 0 = 1/2 -4/4 +2/4
%C E(5) = 1/2 = 1/2 -5/4 +5/4
%C E(6) = 0 = 1/2 -6/4 +9/4 -10/8
%C E(7) = -17/8 = 1/2 -7/4 +14/4 -35/8
%C E(8) = 0 = 1/2 -8/4 +20/4 -80/8 +26/4.
%C The terms of the sequence are the reduced numerators. Like A192456(n) for Bernoulli numbers A164555(n)/A027642(n).
%e a(n) by triangle
%e 1,
%e 1,
%e 1, -2,
%e 1, -3,
%e 1, -4, 2,
%e 1, -5, 5,
%e 1, -6, 9, -10,
%e 1, -7, 14, -35,
%e 1, -8, 20, -80, 26,
%e etc.
%K sign,tabf
%O 0,4
%A _Paul Curtz_, Mar 05 2014