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%I #4 Mar 30 2012 17:34:22
%S 1,61,3070,150836,7392650,362245994,17750074048,869753690956,
%T 42617931038803,2088278621406591,102325652450274784,
%U 5013956970066973919,245683891533290673468,12038510685131268747080,589887023571432406862284
%N G.f. = 1/(x^36*p(1/x)) where p(x)=(- 25 - 49 x^9 + x^10)*(- 1 - 2 x^9 + x^10)^3*(- 1 - x^9 + x^10)^6.
%C Weighted solution of a zero sum game.
%C Let Ma={{0, 1, 0, 0, 0, 0, 0, 0, 0, 0},
%C {0, 0, 1, 0, 0, 0, 0, 0, 0, 0},
%C {0, 0, 0, 1, 0, 0, 0, 0, 0, 0},
%C {0, 0, 0, 0, 1, 0, 0, 0, 0, 0},
%C {0, 0, 0, 0, 0, 1, 0, 0, 0, 0},
%C {0, 0, 0, 0, 0, 0, 1, 0, 0, 0},
%C {0, 0, 0, 0, 0, 0, 0, 1, 0, 0},
%C {0, 0, 0, 0, 0, 0, 0, 0, 1, 0},
%C {0, 0, 0, 0, 0, 0, 0, 0, 0, 1},
%C {25, 0, 0, 0, 0, 0, 0, 0, 0, 49}}; a={1,2};
%C ML={{0, 1, 0, 0, 0, 0, 0, 0, 0, 0},
%C {0, 0, 1, 0, 0, 0, 0, 0, 0, 0},
%C {0, 0, 0, 1, 0, 0, 0, 0, 0, 0},
%C {0, 0, 0, 0, 1, 0, 0, 0, 0, 0},
%C {0, 0, 0, 0, 0, 1, 0, 0, 0, 0},
%C {0, 0, 0, 0, 0, 0, 1, 0, 0, 0},
%C {0, 0, 0, 0, 0, 0, 0, 1, 0, 0},
%C {0, 0, 0, 0, 0, 0, 0, 0, 1, 0},
%C {0, 0, 0, 0, 0, 0, 0, 0, 0, 1},
%C {25, 0, 0, 0, 0, 0, 0, 0, 0, 49}}.
%C Such that:
%C 6*Game_value[M1]+3*Game_value[M2]+Game_Value[ML]=0
%C My first solution was "unweighted".
%F p(x)=(-25 - 49 x^9 + x^10)(-1 - 2 x^9 + x^10)^3(-1 - x^9 + x^10)^6; f(x)=1/(x^36*p(1/x)) a(n) =expansion(f(x))
%t f[x_] = Product[CharacteristicPolynomial[{{0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 0, 0, 0, 0, 0, 0, 0, 0, a}}, x]^(6/a), {a, 1, 2}]*CharacteristicPolynomial[{{0, 1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {25, 0, 0, 0, 0, 0, 0, 0, 0, 49}}, x]; g[x_] = Expand[x^100*f[1/x]]; a = Table[ SeriesCoefficient[Series[1/g[x], {x, 0, 30}], n], {n, 0, 30}]
%K nonn,uned,obsc
%O 1,2
%A _Roger L. Bagula_, Jan 31 2008
%E The connection with the zero-sum game is not clear to me. Also, how does Ma depend on a? It appears that Ma = ML, so perhaps there are errors in these matrices? - _N. J. A. Sloane_, May 16 2008