A115639 -id:A115639 - cp's OEIS frontend

A115636 A divide-and-conquer number triangle.

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, 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, 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

Offset: 0

Paul Barry, Jan 27 2006

			Triangle begins
   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,  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,  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;
  64,   0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1;
  64,   0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1, -1;
  64,   0, 16,  0,  0,  0,  4, -4,  0,  0,  0,  0,  0,  0,  0,  0,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

A115633[n_, k_]:= If[k==n, (-1)^n, If[k==n-1, Mod[n,2], If[n==2*k+2, -4, 0]]];
T[n_, k_]:= T[n, k]= (-1)^k*If[k==n, 1, -Sum[T[n, j]*A115633[j, k], {j,k+1,n}] ];
Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 24 2021 *)

@CachedFunction
def A115633(n, k):
    if (k==n): return (-1)^n
    elif (k==n-1): return n%2
    elif (n==2*k+2): return -4
    else: return 0
def A115636(n,k):
    if (k==0): return 4^(floor(log(n+2, 2)) -1)
    elif (k==n): return (-1)^n
    elif (k==n-1): return (n%2)
    else: return (-1)^(k+1)*sum( A115636(n, j)*A115633(j, k) for j in (k+1..n) )
flatten([[A115636(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Nov 24 2021

T(n, 0) = A115639(n).
Sum_{k=0..n} T(n, k) = A115637(n).
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

A115715 A divide-and-conquer triangle.

Original entry on oeis.org

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, 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, 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

Offset: 0

Paul Barry, Jan 29 2006

			Triangle begins
   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,  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,  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;
  64,  0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1;
  64,  0, 16,  0,  0,  0,  4,  0,  0,  0,  0,  0,  0,  0,  1,  1;
  64,  0, 16,  0,  0,  0,  4,  4,  0,  0,  0,  0,  0,  0,  0,  0,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A032925 (row sums), A115639 (first column), A115713 (inverse).

A115715 := proc(n,k)
    option remember;
    if n = k then
        1;
    elif k > n then
        0;
    else
        -add(procname(n,l)*A115713(l,k),l=k+1..n) ;
    end if;
end proc:
seq(seq(A115715(n,k),k=0..n),n=0..13) ; # R. J. Mathar, Sep 07 2016

A115713[n_, k_]:= If[k==n, 1, If[k==n-1, ((-1)^n-1)/2, If[n==2*k+2, -4, 0]]];
T[n_, k_]:= T[n, k]= If[k==n, 1, -Sum[T[n, j]*A115713[j, k], {j, k+1, n}]];
Table[T[n, k], {n,0,18}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 23 2021 *)

@CachedFunction
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
def A115715(n,k):
    if (k==0): return 4^(floor(log(n+2, 2)) -1)
    elif (k==n): return 1
    elif (k==n-1): return (n%2)
    else: return (-1)*sum( A115715(n,j)*A115713(j,k) for j in (k+1..n) )
flatten([[A115715(n,k) for k in (0..n)] for n in (0..18)]) # G. C. Greubel, Nov 23 2021

Sum_{=0..n} T(n, k) = A032925(n).
T(n, 0) = A115639(n).
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

cp's OEIS Frontend

A115636 A divide-and-conquer number triangle.

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