A178112 Number triangle T(n,k)=C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 3, 0, 1, 0, 1, 0, 3, 0, 3, 0, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 0, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 0, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 1, 0, 6, 0, 15, 0, 20, 0, 15, 0, 6, 0, 1

Offset: 0

Author

Paul Barry, May 20 2010

Keywords

• easy
• nonn
• tabl

Comments

Coefficient array of polynomials P(n,x)=xP(n-1,x)+((1+(-1)^n)/2)*P(n-2,x), P(0,x)=1,P(1,x)=x.

Inverse is A178111.

Examples

			Triangle begins
1,
0, 1,
1, 0, 1,
0, 1, 0, 1,
1, 0, 2, 0, 1,
0, 1, 0, 2, 0, 1,
1, 0, 3, 0, 3, 0, 1,
0, 1, 0, 3, 0, 3, 0, 1,
1, 0, 4, 0, 6, 0, 4, 0, 1,
0, 1, 0, 4, 0, 6, 0, 4, 0, 1,
1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1
Production matrix is
0, 1,
1, 0, 1,
0, 0, 0, 1,
0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Production matrix of inverse is
0, 1,
-1, 0, 1,
0, 0, 0, 1,
0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
• Johann Cigler, Some remarks on the power product expansion of the q-exponential series, arXiv:2006.06242 [math.CO], 2020.

Programs

Maple

A178112 := proc(n,k)
    binomial(floor(n/2),floor(k/2))*( 1+(-1)^(n-k) )/2 ;
end proc:
seq(seq(A178112(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 10 2015

Mathematica

Table[Binomial[Floor[n/2], Floor[k/2]]*(1 + (-1)^(n - k))/2, {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 31 2020 *)