A115636 - OEIS

%I #6 Nov 24 2021 10:40:54

%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,

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

%U 0,0,0,0,1,16,-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 number triangle.

%H G. C. Greubel, <a href="/A115636/b115636.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, 0) = A115639(n).

%F Sum_{k=0..n} T(n, k) = A115637(n).

%F T(n, k) = (-1)^k*( 1 if k = n otherwise (-1)*Sum_{j=k+1..n} T(n, j)*A115633(j, k) ). - _G. C. Greubel_, Nov 24 2021

%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;

%t A115633[n_, k_]:= If[k==n, (-1)^n, If[k==n-1, Mod[n,2], If[n==2*k+2, -4, 0]]];

%t T[n_, k_]:= T[n, k]= (-1)^k*If[k==n, 1, -Sum[T[n, j]*A115633[j, k], {j,k+1,n}] ];

%t Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 24 2021 *)

%o (Sage)

%o @CachedFunction

%o def A115633(n, k):

%o     if (k==n): return (-1)^n

%o     elif (k==n-1): return n%2

%o     elif (n==2*k+2): return -4

%o     else: return 0

%o def A115636(n,k):

%o     if (k==0): return 4^(floor(log(n+2, 2)) -1)

%o     elif (k==n): return (-1)^n

%o     elif (k==n-1): return (n%2)

%o     else: return (-1)^(k+1)*sum( A115636(n, j)*A115633(j, k) for j in (k+1..n) )

%o flatten([[A115636(n, k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Nov 24 2021

%Y Cf. A115633 (inverse), A115637 (row sums), A115639 (first column).

%K sign,tabl

%O 0,4

%A _Paul Barry_, Jan 27 2006