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%I #9 Nov 24 2021 10:40:39
%S 1,1,1,4,0,1,4,0,1,1,4,4,0,0,1,4,4,0,0,1,1,16,0,4,0,0,0,1,16,0,4,0,0,
%T 0,1,1,16,0,4,4,0,0,0,0,1,16,0,4,4,0,0,0,0,1,1,16,16,0,0,4,0,0,0,0,0,
%U 1,16,16,0,0,4,0,0,0,0,0,1,1,16,16,0,0,4,4,0,0,0,0,0,0,1,16,16,0,0,4,4,0,0,0,0,0,0,1,1
%N A divide-and-conquer triangle.
%H G. C. Greubel, <a href="/A115715/b115715.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Sum_{=0..n} T(n, k) = A032925(n).
%F T(n, 0) = A115639(n).
%F T(n, k) = 1 if n = k, otherwise T(n, k) = (-1)*Sum_{j=k+1..n} T(n, j)*A115713(j, k). - _R. J. Mathar_, Sep 07 2016
%e Triangle begins
%e 1;
%e 1, 1;
%e 4, 0, 1;
%e 4, 0, 1, 1;
%e 4, 4, 0, 0, 1;
%e 4, 4, 0, 0, 1, 1;
%e 16, 0, 4, 0, 0, 0, 1;
%e 16, 0, 4, 0, 0, 0, 1, 1;
%e 16, 0, 4, 4, 0, 0, 0, 0, 1;
%e 16, 0, 4, 4, 0, 0, 0, 0, 1, 1;
%e 16, 16, 0, 0, 4, 0, 0, 0, 0, 0, 1;
%e 16, 16, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1;
%e 16, 16, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 1;
%e 16, 16, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 1, 1;
%e 64, 0, 16, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1;
%e 64, 0, 16, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 1;
%e 64, 0, 16, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%p A115715 := proc(n,k)
%p option remember;
%p if n = k then
%p 1;
%p elif k > n then
%p 0;
%p else
%p -add(procname(n,l)*A115713(l,k),l=k+1..n) ;
%p end if;
%p end proc:
%p seq(seq(A115715(n,k),k=0..n),n=0..13) ; # _R. J. Mathar_, Sep 07 2016
%t A115713[n_, k_]:= If[k==n, 1, If[k==n-1, ((-1)^n-1)/2, If[n==2*k+2, -4, 0]]];
%t T[n_, k_]:= T[n, k]= If[k==n, 1, -Sum[T[n, j]*A115713[j, k], {j, k+1, n}]];
%t Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 23 2021 *)
%o (Sage)
%o @CachedFunction
%o def A115713(n,k):
%o if (k==n): return 1
%o elif (k==n-1): return -(n%2)
%o elif (n==2*k+2): return -4
%o else: return 0
%o def A115715(n,k):
%o if (k==0): return 4^(floor(log(n+2, 2)) -1)
%o elif (k==n): return 1
%o elif (k==n-1): return (n%2)
%o else: return (-1)*sum( A115715(n,j)*A115713(j,k) for j in (k+1..n) )
%o flatten([[A115715(n,k) for k in (0..n)] for n in (0..18)]) # _G. C. Greubel_, Nov 23 2021
%Y Cf. A032925 (row sums), A115639 (first column), A115713 (inverse).
%K easy,nonn,tabl
%O 0,4
%A _Paul Barry_, Jan 29 2006
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