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%I #2 Mar 30 2012 17:34:40
%S 1,2,1,4,5,1,8,18,9,1,16,54,51,14,1,32,140,220,115,20,1,64,328,750,
%T 685,225,27,1,128,784,2044,3080,1785,399,35,1,256,2096,5068,10220,
%U 10465,4088,658,44,1,512,4704,16776,25284,43806,30681,8484,1026,54,1,1024
%N A triangle of polynomial coefficients:p(x,n)=Sum[(k + 1)^n*Binomial[x, k], {k, 0, Infinity}]/2^(x - n)
%C Row sums are:A007582;
%C {1, 3, 10, 36, 136, 528, 2080, 8256, 32896, 131328, 524800,...}.
%F p(x,n)=Sum[(k + 1)^n*Binomial[x, k], {k, 0, Infinity}]/2^(x - n);
%F t(n,m)=coefficients(p(x,n))
%e {1},
%e {2, 1},
%e {4, 5, 1},
%e {8, 18, 9, 1},
%e {16, 54, 51, 14, 1},
%e {32, 140, 220, 115, 20, 1},
%e {64, 328, 750, 685, 225, 27, 1},
%e {128, 784, 2044, 3080, 1785, 399, 35, 1},
%e {256, 2096, 5068, 10220, 10465, 4088, 658, 44, 1},
%e {512, 4704, 16776, 25284, 43806, 30681, 8484, 1026, 54, 1},
%e {1024, 2496, 61920, 79980, 118020, 163569, 79905, 16290, 1530, 65, 1}
%t Clear[p, x, n]
%t p[x_, n_] = Sum[(k + 1)^n*Binomial[x, k], {k, 0, Infinity}]/2^(x - n);
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
%t Flatten[%]
%Y Cf. A007582
%K nonn,tabl,uned
%O 0,2
%A _Roger L. Bagula_, Apr 23 2010
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