%I #2 Mar 30 2012 17:34:34
%S 1,126,126,312,882,312,630,3276,3276,630,1116,8820,16224,8820,1116,
%T 1806,19530,54600,54600,19530,1806,2736,37926,145080,220500,145080,
%U 37926,2736,3942,67032,328692,683550,683550,328692,67032,3942,5460,110376
%N Symmetrical Hahn weights on q-form factorials:m=2;q=3; 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, 252, 1506, 7812, 36096, 151872, 591984, 2166432, 7521024, 24970752,
%C 79801344,...}.
%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=2;q=3;
%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 {126, 126},
%e {312, 882, 312},
%e {630, 3276, 3276, 630},
%e {1116, 8820, 16224, 8820, 1116},
%e {1806, 19530, 54600, 54600, 19530, 1806},
%e {2736, 37926, 145080, 220500, 145080, 37926, 2736},
%e {3942, 67032, 328692, 683550, 683550, 328692, 67032, 3942},
%e {5460, 110376, 663936, 1769880, 2421720, 1769880, 663936, 110376, 5460},
%e {7326, 171990, 1229904, 4021920, 7054236, 7054236, 4021920, 1229904, 171990, 7326},
%e {9576, 256410, 2129400, 8278200, 17811360, 22831452, 17811360, 8278200, 2129400, 256410, 9576}
%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,tabl,uned
%O 0,2
%A _Roger L. Bagula_, Feb 26 2009