A115717 - OEIS

%I #10 Nov 24 2021 11:45:16

%S 1,0,1,3,-1,1,0,0,0,1,0,4,-1,-1,1,0,0,0,0,0,1,12,-4,4,0,-1,-1,1,0,0,0,

%T 0,0,0,0,1,0,0,0,4,0,0,-1,-1,1,0,0,0,0,0,0,0,0,0,1,0,16,-4,-4,4,0,0,0,

%U -1,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,-1,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,48,-16,16,0,-4,-4,4,0,0,0,0,0,-1,-1,1

%N A divide-and-conquer triangle related to A007583.

%C Product of (1-x, x), which is A167374, and number triangle A115715.

%H G. C. Greubel, <a href="/A115717/b115717.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Sum_{k=0..n} T(n, k) = A115716(n).

%F T(n ,k) = Sum_{j=k..n} A167374(n, j)*A115715(j, k). - _R. J. Mathar_, Sep 07 2016

%e Triangle begins

%e    1;

%e    0,   1;

%e    3,  -1,  1;

%e    0,   0,  0,  1;

%e    0,   4, -1, -1,  1;

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

%e   12,  -4,  4,  0, -1, -1,  1;

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

%e    0,   0,  0,  4,  0,  0, -1, -1,  1;

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

%e    0,  16, -4, -4,  4,  0,  0,  0, -1, -1,  1;

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

%e    0,   0,  0,  0,  0,  4,  0,  0,  0,  0, -1, -1,  1;

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

%e   48, -16, 16,  0, -4, -4,  4,  0,  0,  0,  0,  0, -1, -1,  1;

%p A115717 := proc(n,k)

%p     add( A167374(n,j)*A115715(j,k),j=k..n) ;

%p end proc: # _R. J. Mathar_, Sep 07 2016

%t A167374[n_, k_]:= If[k>n-2, (-1)^(n-k), 0];

%t g[n_, k_]:= g[n, k]= If[k==n, 1, If[k==n-1, -Mod[n, 2], If[n==2*k+2, -4, 0]]]; (* g = A115713 *)

%t f[n_, k_]:= f[n, k]= If[k==n, 1, -Sum[f[n,j]*g[j,k], {j,k+1,n}]]; (* f=A115715 *)

%t A115717[n_, k_]:= A115717[n, k]= Sum[A167374[n,j]*f[j,k], {j,k,n}];

%t Table[A115717[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 23 2021 *)

%o (Sage)

%o @cached_function

%o def A115717(n,k):

%o     def A167374(n, k):

%o         if (k>n-2): return (-1)^(n-k)

%o         else: return 0

%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+k+1)*A115713(j+k+1,k) for j in (0..n-k-1) )

%o     return sum( A167374(n, j+k)*A115715(j+k, k) for j in (0..n-k) )

%o flatten([[A115717(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Nov 23 2021

%Y Cf. A007583, A115715, A115716 (row sums), A167374.

%K easy,sign,tabl

%O 0,4

%A _Paul Barry_, Jan 29 2006