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%I #9 Dec 25 2013 02:37:54
%S 1,0,1,-1,1,-1,3,-3,-11,11,113,-113,-1269,1269,20575,-20575,-888419,
%T 888419,24729909,-24729909,-862992415,862992415,36913939753,
%U -36913939753,-1899853421901,1899853421901,115841483491307,-115841483491307
%N Numerators of sum of rows of the inverse of the triangle of Euler polynomial coefficients P(0,x)=1, 2P(n,x)=(1+x)*[(1+x)^(n-1)+x^(n-1)].
%C See A133135.
%C Denominators are 1, 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, ..., a sequence which matches A053644, except the first term.
%e 1, 0, 1/2, -1/2, 1/4, -1/4, 3/4, -3/4, -11/8, 11/8, 113/8, -113/8, ...
%t max = 30; p[0, _] = 1; p[n_, x_] := (1+x)*((1+x)^(n-1)+x^(n-1))/2; t = Total /@ Inverse @ Table[Coefficient[p[n, x], x, k], {n, 0, max+2}, {k, 0, max+2}]; a[n_] := t[[n+1]] // Numerator; Table[a[n], {n, 0, max}]
%Y Cf. A053644, A133135.
%K sign,frac,easy
%O 0,7
%A _Jean-François Alcover_ and _Paul Curtz_, Dec 20 2013