A177694 Triangle t(n,m) = p(n)/ (p(m)*p(n-m) ) read by rows, where p(n>=1) = 1, -1, 2, 10, 10, -160, -2080,.. are partial products of A106852.

1, 1, 1, 1, -1, 1, 1, -2, -2, 1, 1, 5, 10, 5, 1, 1, 1, -5, -5, 1, 1, 1, -16, 16, -40, 16, -16, 1, 1, 13, 208, -104, -104, 208, 13, 1, 1, 35, -455, -3640, -728, -3640, -455, 35, 1, 1, -74, 2590, -16835, 53872, 53872, -16835, 2590, -74, 1, 1, -31, -2294, 40145, 104377

Offset: 0

Author

Roger L. Bagula, May 11 2010

Keywords

• sign
• tabl

Comments

Row sums are: 1, 2, 1, -2, 22, -6, -38, 236, -8846, 79108, -1385636,...

Examples

			1;
1, 1;
1, -1, 1;
1, -2, -2, 1;
1, 5, 10, 5, 1;
1, 1, -5, -5, 1, 1;
1, -16, 16, -40, 16, -16, 1;
1, 13, 208, -104, -104, 208, 13, 1;
1, 35, -455, -3640, -728, -3640, -455, 35, 1;
1, -74, 2590, -16835, 53872, 53872, -16835, 2590, -74, 1;
1, -31, -2294, 40145, 104377, -1670032, 104377, 40145, -2294, -31, 1;
		

References

• Advanced Number Theory, Harvey Cohn, Dover Books,1963, Page 47ff

Programs

Mathematica

Clear[t, n, m, c, q, w, f, a];
f[0, a_] := 0; f[1, a_] := 1;
f[n_, a_] := f[n, a] = -f[n - 1, a] - a*f[n - 2, a];
c[n_, a_] := If[n == 0, 1, Product[f[i, a], {i, 1, n}]];
w[n_, m_, q_] := c[n, q]/(c[m, q]*c[n - m, q]);
Table[Flatten[Table[Table[w[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]

Formula

p(n+1) = product_{i=0..n} (-1)^i*A106852(i).

t(n,m) = p(n)/(p(m)*p(n-m)).