OFFSET
0,2
COMMENTS
Row sums are:
{1, 252, 1506, 7812, 36096, 151872, 591984, 2166432, 7521024, 24970752,
79801344,...}.
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.
FORMULA
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])].
EXAMPLE
{1},
{126, 126},
{312, 882, 312},
{630, 3276, 3276, 630},
{1116, 8820, 16224, 8820, 1116},
{1806, 19530, 54600, 54600, 19530, 1806},
{2736, 37926, 145080, 220500, 145080, 37926, 2736},
{3942, 67032, 328692, 683550, 683550, 328692, 67032, 3942},
{5460, 110376, 663936, 1769880, 2421720, 1769880, 663936, 110376, 5460},
{7326, 171990, 1229904, 4021920, 7054236, 7054236, 4021920, 1229904, 171990, 7326},
{9576, 256410, 2129400, 8278200, 17811360, 22831452, 17811360, 8278200, 2129400, 256410, 9576}
MATHEMATICA
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}]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Feb 26 2009
STATUS
approved