A176564 Triangle T(n,m)= binomial(2*n,m) + binomial(2*n,n-m) -binomial(2*n,n) read by rows.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, -6, -14, -6, 1, 1, -32, -87, -87, -32, 1, 1, -120, -363, -484, -363, -120, 1, 1, -415, -1339, -2067, -2067, -1339, -415, 1, 1, -1414, -4742, -7942, -9230, -7942, -4742, -1414, 1, 1, -4844, -16643, -29240, -36992, -36992, -29240

Offset: 0

Author

Roger L. Bagula, Apr 20 2010

Keywords

• sign
• tabl
• easy

Comments

Row sums are 1, 2, 4, 4, -24, -236, -1448, -7640, -37424, -175436,... = 2*A032443(n) -A037965(n+1).

Examples

			The triangle starts in row n=0 with columns 0<=m<=n as:
1;
1, 1;
1, 2, 1;
1, 1, 1, 1;
1, -6, -14, -6, 1;
1, -32, -87, -87, -32, 1;
1, -120, -363, -484, -363, -120, 1;
1, -415, -1339, -2067, -2067, -1339, -415, 1;
1, -1414, -4742, -7942, -9230, -7942, -4742, -1414, 1;
1, -4844, -16643, -29240, -36992, -36992, -29240, -16643, -4844, 1;
1, -16776, -58596, -106096, -141151, -153748, -141151, -106096, -58596, -16776, 1;
		

Programs

Maple

A176564 := proc(n,m) binomial(2*n,m)+binomial(2*n,n-m) -binomial(2*n,n) ; end proc:

Mathematica

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

Formula

T(n,m) = T(n,n-m).