A119328 Number triangle T(n,k)=sum{i=0..n, (-1)^(n-i)*C(n,i)*sum{j=0..i-k, C(k,2j)*C(i-k,2j)}}.

1, 0, 1, 0, -1, 1, 0, 1, -2, 1, 0, -1, 4, -3, 1, 0, 1, -6, 9, -4, 1, 0, -1, 8, -19, 16, -5, 1, 0, 1, -10, 33, -44, 25, -6, 1, 0, -1, 12, -51, 96, -85, 36, -7, 1, 0, 1, -14, 73, -180, 225, -146, 49, -8, 1, 0, -1, 16, -99, 304, -501, 456, -231, 64, -9, 1

Offset: 0

Author

Paul Barry, May 14 2006

Keywords

• easy
• sign
• tabl

Comments

Row sums are A021913(n+2). Product with Pascal's triangle A007318 is A119326.

Examples

			Triangle begins
1,
0, 1,
0, -1, 1,
0, 1, -2, 1,
0, -1, 4, -3, 1,
0, 1, -6, 9, -4, 1,
0, -1, 8, -19, 16, -5, 1,
0, 1, -10, 33, -44, 25, -6, 1,
0, -1, 12, -51, 96, -85, 36, -7, 1,
0, 1, -14, 73, -180, 225, -146, 49, -8, 1,
0, -1, 16, -99, 304, -501, 456, -231, 64, -9, 1
		

Programs

Mathematica

t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n, i]*Sum[Binomial[k, 2 j]*Binomial[i - k, 2 j], {j, 0, i - k}], {i, 0, n}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 25 2013 *)

Formula

Column k has g.f. (x/(1+x))^k*sum{j=0..k, C(k,2j)x^(2j)}