A156184 A generalized recursion triangle sequence : m=1; t(n,k)=(k + m - 1)*t(n - 1, k, m) + (m*n - k + 1 - m)*t(n - 1, k - 1, m).

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 16, 7, 1, 1, 11, 53, 53, 11, 1, 1, 16, 150, 318, 150, 16, 1, 1, 22, 380, 1554, 1554, 380, 22, 1, 1, 29, 892, 6562, 12432, 6562, 892, 29, 1, 1, 37, 1987, 25038, 82538, 82538, 25038, 1987, 37, 1, 1, 46, 4270, 89023, 480380, 825380, 480380

Offset: 0

Author

Roger L. Bagula, Feb 05 2009

Keywords

• nonn
• tabl
• uned

Comments

Row sums are: A054091;

{1, 2, 4, 10, 32, 130, 652, 3914, 27400, 219202, 1972820, ...}.

The sequence comes from a generalization of the recurrence for A008517.

Examples

			{1},
{1, 1},
{1, 2, 1},
{1, 4, 4, 1},
{1, 7, 16, 7, 1},
{1, 11, 53, 53, 11, 1},
{1, 16, 150, 318, 150, 16, 1},
{1, 22, 380, 1554, 1554, 380, 22, 1},
{1, 29, 892, 6562, 12432, 6562, 892, 29, 1},
{1, 37, 1987, 25038, 82538, 82538, 25038, 1987, 37, 1},
{1, 46, 4270, 89023, 480380, 825380, 480380, 89023, 4270, 46, 1}
		

Links

• Eric Weisstein's World of Mathematics, Second-Order Eulerian Triangle.

Crossrefs

Cf. A054091, A054090, A008517, A156141.

Programs

Mathematica

m = 1; e[n_, 0, m_] := 1;
e[n_, k_, m_] := 0 /; k >= n;
e[n_, k_, 1] := 1 /; k >= n;
e[n_, k_, m_] := (k + m - 1)e[n - 1, k, m] + (m*n - k + 1 - m)e[n - 1, k - 1, m];
Table[Table[e[n, k, m], {k, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

t(n,k) = (k + m - 1)*t(n - 1, k, m) + (m*n - k + 1 - m)*t(n - 1, k - 1, m).