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A157320 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])]. 0

%I

%S 1,6,6,8,18,8,10,36,36,10,12,60,96,60,12,14,90,200,200,90,14,16,126,

%T 360,500,360,126,16,18,168,588,1050,1050,588,168,18,20,216,896,1960,

%U 2520,1960,896,216,20,22,270,1296,3360,5292,5292,3360,1296,270,22,24,330

%N 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])].

%C Row sums are:

%C {1, 12, 34, 92, 240, 608, 1504, 3648, 8704, 20480, 47616,...}.

%C These are Rhombi sides as ratios of q-form to factorial:

%C r1=t(1,n)/n!;

%C r2=t(m+1,k]/(n-k)!;

%C r3=t(m+1,n-k)/(n-k)!

%C They get very large very fast, but all are integer.

%F m=1;q=2;

%F q-form:t(n,m)=If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];

%F Hahn weight:

%F b(n,k,m)=If[n == 0, 1, (n!*t[m + 1, k]*t[m + 1, n - k])/(k!*(n - k)!*t[1, n])].

%e {1},

%e {6, 6},

%e {8, 18, 8},

%e {10, 36, 36, 10},

%e {12, 60, 96, 60, 12},

%e {14, 90, 200, 200, 90, 14},

%e {16, 126, 360, 500, 360, 126, 16},

%e {18, 168, 588, 1050, 1050, 588, 168, 18},

%e {20, 216, 896, 1960, 2520, 1960, 896, 216, 20},

%e {22, 270, 1296, 3360, 5292, 5292, 3360, 1296, 270, 22},

%e {24, 330, 1800, 5400, 10080, 12348, 10080, 5400, 1800, 330, 24}

%t Clear[t, n, m, i, k, a, b];

%t t[n_, m_] = If[m == 0, n!, Product[Sum[(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];

%t b[n_, k_, m_] = If[n == 0, 1, (n!*t[m + 1, k]*t[m + 1, n - k])/(k!*(n - k)!*t[ 1, n])];

%t Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

%K nonn,tabf,uned

%O 0,2

%A _Roger L. Bagula_, Feb 26 2009

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