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%I #2 Mar 30 2012 17:34:27
%S 1,1,1,1,3,1,1,7,7,1,1,6,9,6,1,1,11,15,15,11,1,1,9,31,30,31,9,1,1,15,
%T 43,71,71,43,15,1,1,12,42,84,105,84,42,12,1,1,19,54,126,189,189,126,
%U 54,19,1,1,15,91,180,315,378,315,180,91,15,1
%N Symmetrical polynomial: t0(n,m)=If[Mod[2*Binomial[n, m], 2] - Mod[Binomial[n, m], 2] == 0, Binomial[n, m]/2, Binomial[n, m] + 1]; p(x,n)=If[n == 0, 1, (x + 1)^n + Sum[t0(n,m)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]/2].
%C Row sums are:{1, 2, 5, 16, 23, 54, 112, 260, 383, 778, 1582}.
%F t0(n,m)=If[Mod[2*Binomial[n, m], 2] - Mod[Binomial[n, m], 2] == 0, Binomial[n, m]/2, Binomial[n, m] + 1]; p(x,n)=If[n == 0, 1, (x + 1)^n + Sum[t0(n,m)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]/2]; t(n,m)=coefficients(p(x,n)).
%e {1}, {1, 1}, {1, 3, 1}, {1, 7, 7, 1}, {1, 6, 9, 6, 1}, {1, 11, 15, 15, 11, 1}, {1, 9, 31, 30, 31, 9, 1}, {1, 15, 43, 71, 71, 43, 15, 1}, {1, 12, 42, 84, 105, 84, 42, 12, 1}, {1, 19, 54, 126, 189, 189, 126, 54, 19, 1}, {1, 15, 91, 180, 315, 378, 315, 180, 91, 15, 1}
%t Clear[t, p, x, n] t[n_, m_] = If[Mod[2*Binomial[n, m], 2] - Mod[Binomial[n, m], 2] == 0, Binomial[n, m]/2, Binomial[n, m] + 1]; p[x_, n_] = If[n == 0, 1, (x + 1)^n + Sum[t[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]/(2)]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]
%K nonn
%O 0,5
%A _Roger L. Bagula_, Nov 02 2008
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