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])].

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

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..56.

Formula

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])].

Example

{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}

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: A183042 A351516 A083507 * A200616 A019851 A155880

Adjacent sequences: A157317 A157318 A157319 * A157321 A157322 A157323

Keyword

nonn,tabf,uned

Author

Roger L. Bagula, Feb 26 2009

Status

approved