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Number triangle T(n,k)=C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.
3

%I #19 Aug 31 2020 19:40:30

%S 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,

%T 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,

%U 1,0,5,0,10,0,10,0,5,0,1,1,0,6,0,15,0,20,0,15,0,6,0,1

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

%C 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.

%C Inverse is A178111.

%H Michael De Vlieger, <a href="/A178112/b178112.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened.)

%H Johann Cigler, <a href="https://arxiv.org/abs/2006.06242">Some remarks on the power product expansion of the q-exponential series</a>, arXiv:2006.06242 [math.CO], 2020.

%e Triangle begins

%e 1,

%e 0, 1,

%e 1, 0, 1,

%e 0, 1, 0, 1,

%e 1, 0, 2, 0, 1,

%e 0, 1, 0, 2, 0, 1,

%e 1, 0, 3, 0, 3, 0, 1,

%e 0, 1, 0, 3, 0, 3, 0, 1,

%e 1, 0, 4, 0, 6, 0, 4, 0, 1,

%e 0, 1, 0, 4, 0, 6, 0, 4, 0, 1,

%e 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1

%e Production matrix is

%e 0, 1,

%e 1, 0, 1,

%e 0, 0, 0, 1,

%e 0, 0, 1, 0, 1,

%e 0, 0, 0, 0, 0, 1,

%e 0, 0, 0, 0, 1, 0, 1,

%e 0, 0, 0, 0, 0, 0, 0, 1,

%e 0, 0, 0, 0, 0, 0, 1, 0, 1,

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

%e Production matrix of inverse is

%e 0, 1,

%e -1, 0, 1,

%e 0, 0, 0, 1,

%e 0, 0, -1, 0, 1,

%e 0, 0, 0, 0, 0, 1,

%e 0, 0, 0, 0, -1, 0, 1,

%e 0, 0, 0, 0, 0, 0, 0, 1,

%e 0, 0, 0, 0, 0, 0, -1, 0, 1,

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

%p A178112 := proc(n,k)

%p binomial(floor(n/2),floor(k/2))*( 1+(-1)^(n-k) )/2 ;

%p end proc:

%p seq(seq(A178112(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Feb 10 2015

%t 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 *)

%K easy,nonn,tabl

%O 0,13

%A _Paul Barry_, May 20 2010