A165620 - OEIS

%I #12 Jul 16 2019 03:41:02

%S 1,-1,1,0,-1,1,0,-1,-1,1,1,1,-2,-1,1,-1,2,2,-3,-1,1,0,-2,4,3,-4,-1,1,

%T 0,-2,-4,7,4,-5,-1,1,1,2,-6,-7,11,5,-6,-1,1,-1,3,6,-13,-11,16,6,-7,-1,

%U 1,0,-3,9,13,-24,-16,22,7,-8,-1,1

%N Riordan array ((1-x)/(1-x^4),x/(1+x^2)).

%C Diagonal sums are (-1)^n. Row sums have g.f. 1/(1+x^3).

%C The transform of the aerated Catalan numbers by this matrix is (-1)^n.

%C The transform of the shifted central binomial coefficient C(n+1,floor((n+1)/2)) is 1^n.

%C Factorizes as (1/(1+x),x)*(1/(1+x^2),x/(1+x^2)). Inverse is A165621.

%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry4/barry64.html">Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays</a>, JIS 12 (2009) 09.8.6.

%F Number triangle T(n,k)=sum{j=0..n, (-1)^(n-j)(-1)^((j-k)/2)(1+(-1)^(j-k))C((j+k)/2,k)/2}.

%e Triangle begins

%e 1,

%e -1, 1,

%e 0, -1, 1,

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

%e 1, 1, -2, -1, 1,

%e -1, 2, 2, -3, -1, 1,

%e 0, -2, 4, 3, -4, -1, 1,

%e 0, -2, -4, 7, 4, -5, -1, 1,

%e 1, 2, -6, -7, 11, 5, -6, -1, 1,

%e -1, 3, 6, -13, -11, 16, 6, -7, -1, 1

%t (* The function RiordanArray is defined in A256893. *)

%t RiordanArray[(1 - #)/(1 - #^4)&, #/(1 + #^2)&, 11] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)

%K easy,sign,tabl

%O 0,13

%A _Paul Barry_, Sep 22 2009