A140818 - OEIS

%I #11 Nov 04 2018 03:43:10

%S 1,2,2,2,2,2,2,6,6,2,2,8,6,8,2,2,10,20,20,10,2,2,12,30,20,30,12,2,2,

%T 14,42,70,70,42,14,2,2,16,56,112,70,112,56,16,2,2,18,72,168,252,252,

%U 168,72,18,2

%N Coefficients of Hodge diamond binomial 'X' matrices as polynomials: matrix example; M={{1,0,1}. {0,2,0], {1,0,1}: M(d, x, y)= Sum[Sum[If[n == m, Binomial[d - 1, m - 1], If[n == d - m + 1, Binomial[d - 1, n - 1], 0]]*x^(n - 1)*y^(m - 1), {n, 1, d}], {m, 1, d}] .

%C Apparently the same as A139813. - _Georg Fischer_, Nov 02 2018

%F M(d, x, y)= Sum[Sum[If[n == m, Binomial[d - 1, m - 1], If[n == d - m + 1, Binomial[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 {1},

%e {2, 2},

%e {2, 2, 2},

%e {2, 6, 6, 2},

%e {2, 8, 6, 8, 2},

%e {2, 10, 20, 20, 10, 2},

%e {2, 12, 30, 20, 30, 12, 2},

%e {2, 14, 42, 70, 70, 42, 14, 2},

%e {2, 16, 56, 112, 70, 112, 56, 16, 2},

%e {2, 18, 72, 168, 252, 252, 168, 72, 18, 2}.

%t Clear[M, y, x] M[d_, x_, y_] := Sum[Sum[If[n == m, Binomial[d - 1, m - 1], If[n == d - m + 1, Binomial[ 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. A139813, A140685.

%K nonn,tabl,uned

%O 1,2

%A _Roger L. Bagula_ and _Mats Granvik_, Jul 16 2008