%I #3 Oct 13 2012 14:51:33
%S 2,2,4,2,4,6,6,6,6,8,8,12,8,8,10,10,20,20,10,10,12,12,60,60,60,12,12,
%T 14,14,42,70,70,42,14,14,16,16,112,112,280,112,112,16,16,18,18,72,504,
%U 252,252,504,72,18,18
%N Coefficients of Hodge diamond GCD binomial product 'X' matrices as polynomials: matrix example; M={{2,0,2}. {0,2,0], {2,0,2}: M(d, x, y)= Sum[Sum[If[n == m, GCD[d - 1, m - 1], If[n == d - m + 1, GCD[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}] .
%C Row sums are:
%C {0, 4, 10, 24, 44, 80, 228, 280, 792, 1728}.
%F M(d, x, y)=Sum[Sum[If[n == m, Binomial[d - 1, m - 1]* GCD[d - 1, m - 1], If[n == d - m + 1, Binomial[d - 1, n - 1] *GCD[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}] ; a(n,m)=Coefficients(M(n,x,1)).
%e {},
%e {2, 2},
%e 4, 2, 4},
%e {6, 6, 6, 6},
%e {8, 8, 12, 8, 8},
%e {10, 10, 20, 20, 10, 10},
%e {12, 12, 60, 60, 60, 12, 12},
%e {14, 14, 42, 70, 70, 42, 14, 14},
%e 16, 16, 112, 112, 280, 112, 112, 16, 16},
%e {18, 18, 72, 504, 252, 252, 504, 72, 18, 18}
%t Clear[M, y, x] M[d_, x_, y_] := Sum[Sum[If[n == m, Binomial[d - 1, m - 1]* GCD[d - 1, m - 1], If[n == d - m + 1, Binomial[d - 1, n - 1] *GCD[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}]; Table[CoefficientList[M[d, x, 1], x], {d, 1, 10}] Flatten[%] Table[Apply[Plus, CoefficientList[M[d, x, 1], x]], {d, 1, 10}]
%Y Cf. A140685.
%K nonn,uned,tabl
%O 1,1
%A _Roger L. Bagula_ and _Mats Granvik_, Jul 16 2008