A157178 A new general triangle sequence based on the Eulerian form in three parts:m=2; t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]] t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) + (m*k + 1)*t0(n - 1 + 1, k) + m*k*(n - k)*t0(n - 2 + 1, k - 1)].

1, 1, 1, 1, 8, 1, 1, 21, 21, 1, 1, 46, 142, 46, 1, 1, 95, 644, 644, 95, 1, 1, 192, 2439, 5416, 2439, 192, 1, 1, 385, 8415, 34879, 34879, 8415, 385, 1, 1, 770, 27556, 192286, 358454, 192286, 27556, 770, 1, 1, 1539, 87486, 962090, 3001044, 3001044, 962090, 87486

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Feb 24 2009

Keywords

• nonn
• tabl

Comments

Row sums are:

{1, 2, 10, 44, 236, 1480, 10680, 87360, 799680, 8104320, 90115200,...}.

The m=1 of the general sequence is A008518.

Examples

			{1},
{1, 1},
{1, 8, 1},
{1, 21, 21, 1},
{1, 46, 142, 46, 1},
{1, 95, 644, 644, 95, 1},
{1, 192, 2439, 5416, 2439, 192, 1},
{1, 385, 8415, 34879, 34879, 8415, 385, 1},
{1, 770, 27556, 192286, 358454, 192286, 27556, 770, 1},
{1, 1539, 87486, 962090, 3001044, 3001044, 962090, 87486, 1539, 1},
{1, 3076, 272485, 4517480, 21945914, 36637288, 21945914, 4517480, 272485, 3076, 1}
		

Crossrefs

A008518

Programs

Mathematica

Clear[t, n, k, m];
t[n_, k_, m_] = (m*(n - k) + 1)*Binomial[n - 1, k - 1] + (m*k + 1)*Binomial[n - 1, k] - m*k*(n - k)*Binomial[n - 2, k - 1];
Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

Formula

m=2;

t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]];

t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) +

(m*k + 1)*t0(n - 1 + 1, k) +

m*k*(n - k)*t0(n - 2 + 1, k - 1)].