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%I #2 Mar 30 2012 17:34:40
%S 1,1,1,1,-1,1,1,-1,-1,1,1,-10,17,-10,1,1,10,-12,-12,10,1,1,-115,308,
%T -391,308,-115,1,1,599,-1371,769,769,-1371,599,1,1,-4448,11838,-13503,
%U 12219,-13503,11838,-4448,1,1,35864,-97529,102186,-40525,-40525,102186
%N A leading coefficient adjusted symmetrical triangle of polynomial coefficients based on:p(x,n)=Sum[k!*Binomial[x, k], {k, 0, n}]
%C Row sums are:
%C {1, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7,...}.
%F p(x,n)=Sum[k!*Binomial[x, k], {k, 0, n}];
%F t(n,m)=coefficients(p(x,n))+reverse(coefficients(p(x,n)))-1
%e {1},
%e {1, 1},
%e {1, -1, 1},
%e {1, -1, -1, 1},
%e {1, -10, 17, -10, 1},
%e {1, 10, -12, -12, 10, 1},
%e {1, -115, 308, -391, 308, -115, 1},
%e {1, 599, -1371, 769, 769, -1371, 599, 1},
%e {1, -4448, 11838, -13503, 12219, -13503, 11838, -4448, 1},
%e {1, 35864, -97529, 102186, -40525, -40525, 102186, -97529, 35864, 1},
%e {1, -327025, 929363, -1075211, 721544, -497351, 721544, -1075211, 929363, -327025, 1}
%t Clear[p, x, n]
%t p[x_, n_] := Sum[k!*Binomial[x, k], {k, 0, n}];
%t Table[CoefficientList[ExpandAll[p[x, n]], x] + Reverse[CoefficientList[ ExpandAll[p[x, n]], x]] - 1, {n, 0, 10}];
%t Flatten[%]
%K sign,tabl,uned
%O 0,12
%A _Roger L. Bagula_, Apr 23 2010
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