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A157320
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Symmetrical Hahn weights on q-form factorials:m=1;q=2; q-form:t(n,m)=If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; Hahn weight:b(n,k,m)=If[n == 0, 1, (n!*t[m + 1, k]*t[m + 1, n - k])/(k!*(n - k)!*t[1, n])].
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0
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1, 6, 6, 8, 18, 8, 10, 36, 36, 10, 12, 60, 96, 60, 12, 14, 90, 200, 200, 90, 14, 16, 126, 360, 500, 360, 126, 16, 18, 168, 588, 1050, 1050, 588, 168, 18, 20, 216, 896, 1960, 2520, 1960, 896, 216, 20, 22, 270, 1296, 3360, 5292, 5292, 3360, 1296, 270, 22, 24, 330
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OFFSET
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0,2
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COMMENTS
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Row sums are:
{1, 12, 34, 92, 240, 608, 1504, 3648, 8704, 20480, 47616,...}.
These are Rhombi sides as ratios of q-form to factorial:
r1=t(1,n)/n!;
r2=t(m+1,k]/(n-k)!;
r3=t(m+1,n-k)/(n-k)!
They get very large very fast, but all are integer.
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LINKS
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FORMULA
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m=1;q=2;
q-form:t(n,m)=If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
Hahn weight:
b(n,k,m)=If[n == 0, 1, (n!*t[m + 1, k]*t[m + 1, n - k])/(k!*(n - k)!*t[1, n])].
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EXAMPLE
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{1},
{6, 6},
{8, 18, 8},
{10, 36, 36, 10},
{12, 60, 96, 60, 12},
{14, 90, 200, 200, 90, 14},
{16, 126, 360, 500, 360, 126, 16},
{18, 168, 588, 1050, 1050, 588, 168, 18},
{20, 216, 896, 1960, 2520, 1960, 896, 216, 20},
{22, 270, 1296, 3360, 5292, 5292, 3360, 1296, 270, 22},
{24, 330, 1800, 5400, 10080, 12348, 10080, 5400, 1800, 330, 24}
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MATHEMATICA
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Clear[t, n, m, i, k, a, b];
t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, (n!*t[m + 1, k]*t[m + 1, n - k])/(k!*(n - k)!*t[ 1, n])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]
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CROSSREFS
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KEYWORD
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nonn,tabf,uned
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AUTHOR
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STATUS
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approved
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