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A173585 A q-form product triangle based on:q=3;a(n, q)= (Sum[(1 + (-1)^n)*Binomial[n, m]*(1 + Sqrt[q])^m, {m, 1, n}] + Sum[(1 + (-1)^n)*Binomial[n, m]*(1 - Sqrt[q])^m, {m, 1, n}])/4 0
1, 1, 1, 1, 16, 1, 1, 225, 225, 1, 1, 3136, 44100, 3136, 1, 1, 43681, 8561476, 8561476, 43681, 1, 1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1, 1, 8473921, 322220846025, 62555239000969, 62555239000969, 322220846025, 8473921, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, 18, 452, 50374, 17210316, 26473387956, 125754936641832,

2692503748294554438, 178119744099983099364620, 53115099293451187427426853340,...}.

Most of these triangles are rational.

LINKS

Table of n, a(n) for n=0..36.

FORMULA

q=2;a(n, q)=(Sum[(1 + (-1)^n)*Binomial[n, m]*(1 + Sqrt[q])^m, {m, 1, n}] + Sum[(1 + (-1)^n)*Binomial[n, m]*(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, 16, 1},

{1, 225, 225, 1},

{1, 3136, 44100, 3136, 1},

{1, 43681, 8561476, 8561476, 43681, 1},

{1, 608400, 1660970025, 23150231104, 1660970025, 608400, 1},

{1, 8473921, 322220846025, 62555239000969, 62555239000969, 322220846025, 8473921, 1},

{1, 118026496, 62509200188176, 169024877308827904, 2354328975040469284, 169024877308827904, 62509200188176, 118026496, 1},

{1, 1643897025, 12126462852848400, 456705280997653184784, 88603154642529399752100, 88603154642529399752100, 456705280997653184784, 12126462852848400, 1643897025, 1},

{1, 22896531856, 2352471287556008025, 1234017524492640137505024, 3334492032897384440894996964, 46443647187902490644456769600, 3334492032897384440894996964, 1234017524492640137505024, 2352471287556008025, 22896531856, 1}

MATHEMATICA

a[n_, q_] := (Sum[(1 + (-1)^n)*Binomial[n, m]*(1 + Sqrt[q])^m, {m, 1, n}] + Sum[(1 + (-1)^n)*Binomial[n, m]*(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: A142462 A203397 A173885 * A022179 A015141 A176390

Adjacent sequences:  A173582 A173583 A173584 * A173586 A173587 A173588

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Feb 22 2010

STATUS

approved

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Last modified May 29 17:00 EDT 2017. Contains 287253 sequences.