A140818
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}] .
Original entry on oeis.org
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, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 70, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2
Offset: 1
{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, 14, 42, 70, 70, 42, 14, 2},
{2, 16, 56, 112, 70, 112, 56, 16, 2},
{2, 18, 72, 168, 252, 252, 168, 72, 18, 2}.
-
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}];
Flatten@ Table[CoefficientList[M[d, x, 1], x], {d, 1, 10}]
A140819
Triangle read by rows: T(n, m) = m if 2*m = n, otherwise 2*gcd(n, m).
Original entry on oeis.org
0, 2, 2, 4, 1, 4, 6, 2, 2, 6, 8, 2, 2, 2, 8, 10, 2, 2, 2, 2, 10, 12, 2, 4, 3, 4, 2, 12, 14, 2, 2, 2, 2, 2, 2, 14, 16, 2, 4, 2, 4, 2, 4, 2, 16, 18, 2, 2, 6, 2, 2, 6, 2, 2, 18
Offset: 0
{0},
{2, 2},
{4, 1, 4},
{6, 2, 2, 6},
{8, 2, 2, 2, 8},
{10, 2, 2, 2, 2, 10},
{12, 2, 4, 3, 4, 2, 12},
{14, 2, 2, 2, 2, 2, 2, 14},
{16, 2, 4, 2, 4, 2, 4, 2, 16},
{18, 2, 2, 6, 2, 2, 6, 2, 2, 18}
-
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}]
Table[CoefficientList[M[d, x, 1], x], {d, 1, 10}]
Flatten[%]
Table[Apply[Plus, CoefficientList[M[d, x, 1], x]], {d, 1, 10}]
Definition simplified by the editors of the OEIS, Jan 03 2024
A140821
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}] .
Original entry on oeis.org
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, 14, 14, 42, 70, 70, 42, 14, 14, 16, 16, 112, 112, 280, 112, 112, 16, 16, 18, 18, 72, 504, 252, 252, 504, 72, 18, 18
Offset: 1
{},
{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},
{14, 14, 42, 70, 70, 42, 14, 14},
{16, 16, 112, 112, 280, 112, 112, 16, 16},
{18, 18, 72, 504, 252, 252, 504, 72, 18, 18}
-
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, d}], {m, d}];
Flatten@ Table[CoefficientList[M[d, x, 1], x], {d, 10}]
Showing 1-3 of 3 results.
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