cp's OEIS Frontend

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

%I A178112 #19 Aug 31 2020 19:40:30
%S A178112 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 A178112 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 A178112 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 A178112 Number triangle T(n,k)=C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.
%C A178112 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 A178112 Inverse is A178111.
%H A178112 Michael De Vlieger, <a href="/A178112/b178112.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened.)
%H A178112 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 A178112 Triangle begins
%e A178112 1,
%e A178112 0, 1,
%e A178112 1, 0, 1,
%e A178112 0, 1, 0, 1,
%e A178112 1, 0, 2, 0, 1,
%e A178112 0, 1, 0, 2, 0, 1,
%e A178112 1, 0, 3, 0, 3, 0, 1,
%e A178112 0, 1, 0, 3, 0, 3, 0, 1,
%e A178112 1, 0, 4, 0, 6, 0, 4, 0, 1,
%e A178112 0, 1, 0, 4, 0, 6, 0, 4, 0, 1,
%e A178112 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1
%e A178112 Production matrix is
%e A178112 0, 1,
%e A178112 1, 0, 1,
%e A178112 0, 0, 0, 1,
%e A178112 0, 0, 1, 0, 1,
%e A178112 0, 0, 0, 0, 0, 1,
%e A178112 0, 0, 0, 0, 1, 0, 1,
%e A178112 0, 0, 0, 0, 0, 0, 0, 1,
%e A178112 0, 0, 0, 0, 0, 0, 1, 0, 1,
%e A178112 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
%e A178112 Production matrix of inverse is
%e A178112 0, 1,
%e A178112 -1, 0, 1,
%e A178112 0, 0, 0, 1,
%e A178112 0, 0, -1, 0, 1,
%e A178112 0, 0, 0, 0, 0, 1,
%e A178112 0, 0, 0, 0, -1, 0, 1,
%e A178112 0, 0, 0, 0, 0, 0, 0, 1,
%e A178112 0, 0, 0, 0, 0, 0, -1, 0, 1,
%e A178112 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
%p A178112 A178112 := proc(n,k)
%p A178112     binomial(floor(n/2),floor(k/2))*( 1+(-1)^(n-k) )/2 ;
%p A178112 end proc:
%p A178112 seq(seq(A178112(n,k),k=0..n),n=0..10) ; # _R. J. Mathar_, Feb 10 2015
%t A178112 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 A178112 easy,nonn,tabl
%O A178112 0,13
%A A178112 _Paul Barry_, May 20 2010