A173584 - OEIS

%I #2 Mar 30 2012 17:34:39

%S 1,1,1,1,7,1,1,42,42,1,1,246,1476,246,1,1,1435,50430,50430,1435,1,1,

%T 8365,1714825,10043975,1714825,8365,1,1,48756,58263420,1990666850,

%U 1990666850,58263420,48756,1,1,284172,1979298576,394210299720

%N A q-form product triangle based on:q=2;a(n, q)= (Sum[(1 + (-1)^n)*(1 + Sqrt[q])^m, {m, 1, n}] + Sum[(1 + (-1)^n)*(1 - Sqrt[q])^m, {m, 1, n}])/4

%C Row sums are:

%C {1, 2, 9, 86, 1970, 103732, 13490357, 4097958054, 3091939846638,

%C 5464546910806332, 24011812170568362074,...}.

%C Most of these triangles are rational.

%F q=2;a(n, q)=(Sum[(1 + (-1)^n)*(1 + Sqrt[q])^m, {m, 1, n}] + Sum[(1 + (-1)^n)*(1 - Sqrt[q])^m, {m, 1, n}])/4;

%F c(n,q)=Product[a(k, q), {k, 2, n, 2}];

%F t(n,m,q)=c(n, q)/(c(m, q)*c(n - m, q))

%e {1},

%e {1, 1},

%e {1, 7, 1},

%e {1, 42, 42, 1},

%e {1, 246, 1476, 246, 1},

%e {1, 1435, 50430, 50430, 1435, 1},

%e {1, 8365, 1714825, 10043975, 1714825, 8365, 1},

%e {1, 48756, 58263420, 1990666850, 1990666850, 58263420, 48756, 1},

%e {1, 284172, 1979298576, 394210299720, 2299560081700, 394210299720, 1979298576, 284172, 1},

%e {1, 1656277, 67238221092, 78053969227656, 2654152246298140, 2654152246298140, 78053969227656, 67238221092, 1656277, 1},

%e {1, 9653491, 2284122159001, 15454370527800766, 3062980851436805476, 17854937158375524604, 3062980851436805476, 15454370527800766, 2284122159001, 9653491, 1}

%t a[n_, q_] := (Sum[(1 + (-1)^n)*(1 + Sqrt[q])^m, {m, 1, n}] + Sum[(1 + (-1)^n)*(1 - Sqrt[q])^m, {m, 1, n}])/4;

%t c[n_, q_] := Product[a[k, q], {k, 2, n, 2}];

%t t[n_, m_, q_] := c[n, q]/(c[m, q]*c[n - m, q]);

%t Table[Table[Table[t[n, m, q], {m, 0, n, 2}], {n, 0, 20, 2}], {q, 1, 10}];

%t Table[Flatten[ Table[Table[t[n, m, q], {m, 0, n, 2}], {n, 0, 20, 2}]], {q, 1, 10}]

%K nonn,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Feb 22 2010