A176158 Triangle read by rows: T(n,m) = (1 + 2 * binomial(n,m))^n for 0 <= m <= n, n >= 0.

1, 3, 3, 9, 25, 9, 27, 343, 343, 27, 81, 6561, 28561, 6561, 81, 243, 161051, 4084101, 4084101, 161051, 243, 729, 4826809, 887503681, 4750104241, 887503681, 4826809, 729, 2187, 170859375, 271818611107, 9095120158391, 9095120158391, 271818611107, 170859375, 2187

Offset: 0

Author

Roger L. Bagula, Apr 10 2010

Keywords

• nonn
• tabl

Comments

Row sums are: 1, 6, 43, 740, 41845, 8490790, 6534766679, 18734219262120, 209617607911694569, 8719076076193077820874, 1429879617351180068934959131, ... .

Examples

			{1},
{3, 3},
{9, 25, 9},
{27, 343, 343, 27},
{81, 6561, 28561, 6561, 81},
{243, 161051, 4084101, 4084101, 161051, 243},
{729, 4826809, 887503681, 4750104241, 887503681, 4826809, 729},
{2187, 170859375, 271818611107, 9095120158391, 9095120158391, 271818611107, 170859375, 2187}.
		

Crossrefs

Columns m=0-1 give: A000244, A085527.

Programs

Maple

f:= proc(n) local m; seq((binomial(n,m)*2+1)^n, m=0..n) end proc:
for n from 0 to 10 do f(n) od; # Robert Israel, Dec 04 2024

Mathematica

Clear[p, n, m];
p[x_, n_, m_] := (1 + 2*Binomial[n, m]*x)^n;
Table[Table[ Apply[Plus, CoefficientList[p[x, n, m], x]], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Formula

T(n,m) = (1 + 2*binomial(n,m))^n.

Extensions

Edited by Robert Israel, Dec 04 2024