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A115713
A divide-and-conquer related triangle.
5
1, -1, 1, -4, 0, 1, 0, 0, -1, 1, 0, -4, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, -4, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, -4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 1
OFFSET
0,4
FORMULA
G.f.: (1-x+x*y)/(1-x^2*y^2) - 4*x^2/(1-x^2*y).
(1, x) - (x, x)/2 - (x, -x)/2 - 4*(x^2, x^2) expressed in the notation of stretched Riordan arrays.
Column k has g.f.: x^k - (x*(-x)^k + x^(k+1))/2 - 4*x^(2*k+2).
T(n, k) = if(n=k, 1, 0) OR if(n=2k+2, -4, 0) OR if(n=k+1, -(1+(-1)^k)/2, 0).
Sum_{k=0..n} T(n, k) = A115634(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A115714(n).
EXAMPLE
Triangle begins
1;
-1, 1;
-4, 0, 1;
0, 0, -1, 1;
0, -4, 0, 0, 1;
0, 0, 0, 0, -1, 1;
0, 0, -4, 0, 0, 0, 1;
0, 0, 0, 0, 0, 0, -1, 1;
0, 0, 0, -4, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 0, 0, 0, -1, 1;
0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1;
0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 1;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1;
0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 1;
MAPLE
A115713 := proc(n, k)
coeftayl( (1-x+x*y)/(1-x^2*y^2)-4*x^2/(1-x^2*y), x=0, n) ;
coeftayl( %, y=0, k) ;
end proc: # R. J. Mathar, Sep 07 2016
MATHEMATICA
T[n_, k_]:= If[k==n, 1, If[k==n-1, -(1-(-1)^n)/2, If[n==2*k+2, -4, 0]]];
Table[T[n, k], {n, 0, 18}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)
PROG
(Sage)
def A115713(n, k):
if (k==n): return 1
elif (k==n-1): return -(n%2)
elif (n==2*k+2): return -4
else: return 0
flatten([[A115713(n, k) for k in (0..n)] for n in (0..18)]) # G. C. Greubel, Nov 23 2021
CROSSREFS
Cf. A115634 (row sums), A115714 (diagonal sums), A115715 (inverse).
Sequence in context: A321188 A322076 A115633 * A199571 A036859 A036861
KEYWORD
easy,sign,tabl
AUTHOR
Paul Barry, Jan 29 2006
STATUS
approved