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%I #2 Mar 30 2012 17:34:34
%S 2,1,1,1,-1,2,1,3,-7,6,1,-12,40,-46,24,1,60,-260,430,-326,120,1,-360,
%T 1920,-4140,4536,-2556,720,1,2520,-15960,42420,-60732,49644,-22212,
%U 5040,1,-20160,147840,-467040,825216,-883008,574848,-212976,40320,1,181440
%N A triangle of infinite sum coefficients with: Limit[Log[1-x],x->0]=-x: p(x,y)=1+n!*x^(n - 1)*Sum[x^k/(k*Binomial[n + k, k]), {k, 1, Infinity}]; such that Log[1-x]->-x.
%C Row sums are:1+n!;
%C {2, 2, 2, 3, 7, 25, 121, 721, 5041, 40321, 362881,...}.
%F Limit[Log[1-x],x->0]=-x:
%F p(x,y)=1+n!*x^(n - 1)*Sum[x^k/(k*Binomial[n + k, k]), {k, 1, Infinity}];
%F such that Log[1-x]->-x.
%e {2},
%e {1, 1},
%e {1, -1, 2},
%e {1, 3, -7, 6},
%e {1, -12, 40, -46, 24},
%e {1, 60, -260, 430, -326, 120},
%e {1, -360, 1920, -4140, 4536, -2556, 720},
%e {1, 2520, -15960, 42420, -60732, 49644, -22212, 5040},
%e {1, -20160, 147840, -467040, 825216, -883008, 574848, -212976, 40320},
%e {1, 181440, -1512000, 5533920, -11630304, 15374016, -13120704, 7090416, -2239344, 362880},
%e {1, -1814400, 16934400, -70459200, 171642240, -270043200, 284947200, -202111200, 93297600, -25659360, 3628800}
%t Clear[p, x, n, m];
%t p[x_, n_] = n!*x^(n - 1)*Sum[x^k/(k*Binomial[n + k, k]), {k, 1, Infinity}]
%t Table[ExpandAll[1 + p[x, n] /. Log[1 - x] -> -x], {n, 0, 10}]
%t Table[CoefficientList[ExpandAll[1 + p[x, n] /. Log[1 - x] -> - x], x], {n, 0, 10}];
%t Flatten[%]
%K sign,tabl,uned
%O 0,1
%A _Roger L. Bagula_, Feb 22 2009
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