A157322 Symmetrical Hahn weights on q-form factorials:m=3;q=4; 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])].

1, 7560, 7560, 49920, 198450, 49920, 214200, 1965600, 1965600, 214200, 696384, 11245500, 25958400, 11245500, 696384, 1871016, 45700200, 185640000, 185640000, 45700200, 1871016, 4377600, 147342510, 905299200, 1593112500, 905299200

Offset

0,2

Comments

Row sums are:

{1, 15120, 298290, 4359600, 49842168, 466422432, 3707151120, 25761076800,

160081662720, 905384837376, 4726028289024,...}.

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.

Links

Table of n, a(n) for n=0..25.

Formula

m=3;q=4;

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},
{7560, 7560},
{49920, 198450, 49920},
{214200, 1965600, 1965600, 214200},
{696384, 11245500, 25958400, 11245500, 696384},
{1871016, 45700200, 185640000, 185640000, 45700200, 1871016},
{4377600, 147342510, 905299200, 1593112500, 905299200, 147342510, 4377600},
{9224280, 402192000, 3405249120, 9063873000, 9063873000, 3405249120, 402192000, 9224280},
{17908800, 968549400, 10622976000, 38963908200, 58934977920, 38963908200, 10622976000, 968549400, 17908800},
{32556744, 2115477000, 28779753600, 136745280000, 285019351344, 285019351344, 136745280000, 28779753600, 2115477000, 32556744}, {56077056, 4273072650, 69844320000, 411633495000, 1111428864000, 1531556631612, 1111428864000, 411633495000, 69844320000, 4273072650, 56077056}

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

Sequence in context: A031137 A210171 A234987 * A172443 A190108 A308913

Adjacent sequences: A157319 A157320 A157321 * A157323 A157324 A157325

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Feb 26 2009

Status

approved