A173584 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

1, 1, 1, 1, 7, 1, 1, 42, 42, 1, 1, 246, 1476, 246, 1, 1, 1435, 50430, 50430, 1435, 1, 1, 8365, 1714825, 10043975, 1714825, 8365, 1, 1, 48756, 58263420, 1990666850, 1990666850, 58263420, 48756, 1, 1, 284172, 1979298576, 394210299720

Offset

0,5

Comments

Row sums are:

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

5464546910806332, 24011812170568362074,...}.

Most of these triangles are rational.

Links

Table of n, a(n) for n=0..39.

Formula

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

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

Example

{1},
{1, 1},
{1, 7, 1},
{1, 42, 42, 1},
{1, 246, 1476, 246, 1},
{1, 1435, 50430, 50430, 1435, 1},
{1, 8365, 1714825, 10043975, 1714825, 8365, 1},
{1, 48756, 58263420, 1990666850, 1990666850, 58263420, 48756, 1},
{1, 284172, 1979298576, 394210299720, 2299560081700, 394210299720, 1979298576, 284172, 1},
{1, 1656277, 67238221092, 78053969227656, 2654152246298140, 2654152246298140, 78053969227656, 67238221092, 1656277, 1},
{1, 9653491, 2284122159001, 15454370527800766, 3062980851436805476, 17854937158375524604, 3062980851436805476, 15454370527800766, 2284122159001, 9653491, 1}

Mathematica

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[n_, q_] := Product[a[k, q], {k, 2, n, 2}];
t[n_, m_, q_] := c[n, q]/(c[m, q]*c[n - m, q]);
Table[Table[Table[t[n, m, q], {m, 0, n, 2}], {n, 0, 20, 2}], {q, 1, 10}];
Table[Flatten[ Table[Table[t[n, m, q], {m, 0, n, 2}], {n, 0, 20, 2}]], {q, 1, 10}]

Crossrefs

Sequence in context: A033933 A108267 A156916 * A166973 A157156 A022170

Adjacent sequences: A173581 A173582 A173583 * A173585 A173586 A173587

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Feb 22 2010

Status

approved