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

0,13

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.

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.

Example

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

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 *)

Crossrefs

Sequence in context: A325488 A339813 A178111 * A324852 A363854 A353967

Adjacent sequences: A178109 A178110 A178111 * A178113 A178114 A178115

Keyword

easy,nonn,tabl

Author

Paul Barry, May 20 2010

Status

approved