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A178112
Number triangle T(n,k)=C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.
3
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
KEYWORD
easy,nonn,tabl
AUTHOR
Paul Barry, May 20 2010
STATUS
approved