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%I #5 Oct 12 2012 14:54:49
%S 1,2,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,11,11,1,1,1,1,26,66,26,1,1,1,1,57,
%T 302,302,57,1,1,1,1,120,1191,2416,1191,120,1,1,1,1,247,4293,15619,
%U 15619,4293,247,1,1,1,1,502,14608,88234,156190,88234,14608,502,1,1,1,1,1013
%N An extended triangle of Eulerian coefficients A123125: f(x,n)=x^(n+1)+1+A123125(x,n).
%C Not entirely symmetrical, the x^(n+1)+1 polynomials was added to remove zeros and make the triangle more symmetrical.
%C Row sums are:
%C {1, 3, 3, 4, 8, 26, 122, 722, 5042, 40322, 362882, 3628802}.
%F f(x,n)=x^(n+1)+1+A123125(x,n).
%t lear[f, x, n, a] f[x_, n_] := f[x,n] = x^(n + 1) + (1 - x)^(n + 1)*Sum[k^n*x^k, {k, 0, Infinity}] + 1; Table[FullSimplify[ExpandAll[f[x, n]]], {n, 0, 10}]; a = Join[{{1}}, Table[CoefficientList[FullSimplify[ExpandAll[f[x, n]]], x], {n, 0, 10}]]; Flatten[a]
%Y Cf. A123125.
%K nonn,tabl
%O 1,2
%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 25 2008
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