A142158 - OEIS

%I #2 Mar 30 2012 17:34:27

%S 0,1,1,4,-3,1,27,-44,31,-8,256,-655,731,-389,81,3125,-10974,17026,

%T -13934,5901,-1024,46656,-208943,418377,-465898,300182,-105279,15625,

%U 823543,-4491192,11064957,-15661904,13617801,-7229592,2161363,-279936,16777216,-107948223,316559287,-545245307,598756419

%N An infinite sum polynomial triangular sequence of coefficients that gives a LerchPhi polynomial: p(x,n)=(1 - x)^(n + 1)*Sum[(n + k)^n*x^k, {k, 0, Infinity}]=(1+x)^n*LerchPhi[x,-n,n].

%C The row sums are n!.

%F p(x,n)=(1 - x)^(n + 1)*Sum[(n + k)^n*x^k, {k, 0, Infinity}]=(1+x)^n*LerchPhi[x,-n,n]; t(n,m)=coefficients(p(x,n)).

%e {0, 1},

%e {1},

%e {4, -3, 1},

%e {27, -44,31, -8},

%e {256, -655, 731, -389, 81},

%e {3125, -10974, 17026, -13934, 5901, -1024},

%e {46656, -208943, 418377, -465898, 300182, -105279, 15625},

%e {823543, -4491192, 11064957, -15661904,13617801, -7229592, 2161363, -279936},

%e {16777216, -107948223, 316559287, -545245307, 598756419, -427227197, 192806917, -50203593, 5764801},

%e {387420489, -2874204890, 9791869696, -19910155238, 26472644638, -23777517254, 14389038880, -5646339386, 1301823673, -134217728},

%e {10000000000, -84062575399, 326605693613, -766674161560, 1198591217792, -1299948741046, 988352227754, -519310387408, 180244457240, -37280886587, 3486784401}

%t Clear[p, x, n]; p[x_, n_] = (1 - x)^(n + 1)*Sum[(n + k)^n*x^k, {k, 0, Infinity}]; Table[FullSimplify[Expand[p[x, n]]], {n, 0, 10}]; Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 0, 10}]; Flatten[%]

%K sign,uned

%O 1,4

%A _Roger L. Bagula_, Sep 16 2008