A156587 A new q-combination type general triangle sequence based on Stirling first polynomials: here q=5: m=4: t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])].

1, 1, 1, 1, 5, 1, 1, 30, 30, 1, 1, 210, 1260, 210, 1, 1, 1680, 70560, 70560, 1680, 1, 1, 15120, 5080320, 35562240, 5080320, 15120, 1, 1, 151200, 457228800, 25604812800, 25604812800, 457228800, 151200, 1, 1, 1663200, 50295168000, 25348764672000

Offset: 0

Author

Roger L. Bagula, Feb 10 2009

Keywords

• nonn
• tabl
• uned

Comments

Row sums are:

{1, 2, 7, 62, 1682, 144482, 45753122, 52124385602, 253588240382402,

4885227205552108802, 454865349223042267910402,...}.

The q=2 sequence is A009963.

Examples

			{1},
{1, 1},
{1, 5, 1},
{1, 30, 30, 1},
{1, 210, 1260, 210, 1},
{1, 1680, 70560, 70560, 1680, 1},
{1, 15120, 5080320, 35562240, 5080320, 15120, 1},
{1, 151200, 457228800, 25604812800, 25604812800, 457228800, 151200, 1},
{1, 1663200, 50295168000, 25348764672000, 202790117376000, 25348764672000, 50295168000, 1663200, 1},
{1, 19958400, 6638962176000, 33460369367040000, 2409146594426880000, 2409146594426880000, 33460369367040000, 6638962176000, 19958400, 1},
{1, 259459200, 1035678099456000, 57417993833840640000, 41340955560365260800000, 372068600043287347200000, 41340955560365260800000, 57417993833840640000, 1035678099456000, 259459200, 1}
		

Crossrefs

A009963

Programs

Mathematica

Clear[t, n, m, i, k, a, b];
t[n_, m_] = If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];
b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];
Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

Formula

q=5: m=4:

t(n,k)=If[m == 0, n!, Product[Sum[(-1)^(i + k)*StirlingS1[k - 1, i]*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}]];

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