A158856 - OEIS

%I #10 Mar 08 2022 02:03:47

%S 1,1,1,1,0,-1,1,0,0,-1,1,0,1,0,1,1,0,1,1,0,1,1,0,1,0,-1,0,-1,1,0,1,0,

%T 0,-1,0,-1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,1,0,1,0,1,0,-1,0,-1,

%U 0,-1

%N Triangle T(n, k) = coefficients of p(n, x), where p(n, x) = (1 - x^(2+floor((n-1)/2)))*(1 + (-1)^floor(n/2)*x^(1+floor(n/2))), read by rows.

%H G. C. Greubel, <a href="/A158856/b158856.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = coefficients of p(n, x), where p(n, x) = (Sum_{j=0..1+floor((n-1)/2)} x^j)*(Sum_{i=0..floor(n/2)} (-x)^i) and p(0, x) = 1.

%F From _G. C. Greubel_, Mar 07 2022: (Start)

%F T(n, k) = coefficients of p(n, x), where p(n, x) = (1 - x^(2+floor((n-1)/2)))*(1 + (-1)^floor(n/2)*x^(1+floor(n/2))).

%F Sum_{k=0..n} T(n, k) = floor((n+3)/2)*( (1 + floor(n/2)) mod 2 ).

%F Sum_{k=0..n} abs(T(n, k)) = A004524(n+3).

%F T(2*n, n) = (1 + (-1)^n)/2.

%F T(2*n+1, n) = (1 + (-1)^n)/2.

%F Sum_{k=0..floor(n/2)} T(n, k) = floor((n+4)/4).

%F T(n, k) = abs(A154957(n,k)). (End)

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  0, -1;

%e   1,  0,  0, -1;

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

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

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

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

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

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

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

%t p[x_, n_]= (1-x^(2+Floor[(n-1)/2]))*(1+(-1)^Floor[n/2]*x^(1+Floor[n/2]))/(1 - x^2);

%t Table[CoefficientList[p[x, n], x], {n,0,12}]//Flatten (* modified by _G. C. Greubel_, Mar 07 2022 *)

%o (Sage)

%o def p(n,x): return (1-x^(2+((n-1)//2)))*(1+(-1)^(n//2)*x^(1+(n//2)))/(1-x^2)

%o def A158856(n,k): return ( p(n,x) ).series(x, n+1).list()[k]

%o flatten([[A158856(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 07 2022

%Y Cf. A004524, A154957.

%K sign,tabl

%O 0,1

%A _Roger L. Bagula_, Mar 28 2009

%E Edited by _G. C. Greubel_, Mar 07 2022