cp's OEIS Frontend

A115715 A divide-and-conquer triangle.

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