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%I #6 Aug 09 2015 11:59:31
%S 1,1,1,1,4,1,1,9,9,1,1,22,22,22,1,1,61,54,54,61,1,1,190,143,132,143,
%T 190,1,1,647,421,339,339,421,647,1,1,2344,1372,952,838,952,1372,2344,
%U 1,1,8841,4836,2964,2238,2238,2964,4836,8841,1,1,34186,17965,10104,6610
%N A functionally symmetric Polynomial as a triangle of coefficients: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[(2^m + 2*m )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
%C Row sums are:{1, 2, 6, 20, 68, 232, 800, 2816, 10176, 37760, 143360}.
%F p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[(2^m + 2*m )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; t(n,m)=coefficients(p(x,n)).
%F Conjecture: row sums are 2^n*(2^n+6-n+n^2)/8 for n>0. [From _R. J. Mathar_, Nov 30 2008]
%e {1},
%e {1, 1},
%e {1, 4, 1},
%e {1, 9, 9, 1},
%e {1, 22, 22, 22, 1},
%e {1, 61, 54, 54, 61, 1},
%e {1, 190, 143, 132, 143, 190, 1},
%e {1, 647, 421, 339, 339, 421, 647, 1},
%e {1, 2344, 1372, 952, 838, 952, 1372, 2344, 1},
%e {1, 8841, 4836, 2964, 2238, 2238, 2964, 4836, 8841, 1},
%e {1, 34186, 17965, 10104, 6610, 5628, 6610, 10104, 17965, 34186, 1}
%t Clear[p, x, n]; p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[(2^m + 2*m )*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Nov 03 2008
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