A104975 Inverse of a Fredholm-Rueppel triangle.

1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 1, 0, -1, 0, 1, 0, 1, 0, -1, 0, 1, -2, 0, 1, 0, -1, 0, 1, 0, -2, 0, 1, 0, -1, 0, 1, 3, 0, -2, 0, 1, 0, -1, 0, 1, 0, 3, 0, -2, 0, 1, 0, -1, 0, 1, -4, 0, 3, 0, -2, 0, 1, 0, -1, 0, 1, 0, -4, 0, 3, 0, -2, 0, 1, 0, -1, 0, 1, 6, 0, -4, 0, 3, 0, -2, 0, 1, 0, -1, 0, 1, 0, 6, 0, -4, 0, 3, 0, -2, 0, 1, 0, -1, 0, 1, -10, 0, 6, 0, -4, 0, 3, 0, -2

Offset: 0

Author

Paul Barry, Mar 30 2005

Keywords

• easy
• sign
• tabl

Comments

Sequence array for A104977.

Inverse of A104974.

Examples

			Triangle begins as:
   1;
   0,  1;
  -1,  0,  1;
   0, -1,  0,  1;
   1,  0, -1,  0,  1;
   0,  1,  0, -1,  0,  1;
  -2,  0,  1,  0, -1,  0,  1;
   0, -2,  0,  1,  0, -1,  0,  1;
   3,  0, -2,  0,  1,  0, -1,  0,  1;
   0,  3,  0, -2,  0,  1,  0, -1,  0,  1;
  -4,  0,  3,  0, -2,  0,  1,  0, -1,  0,  1;
   0, -4,  0,  3,  0, -2,  0,  1,  0, -1,  0, 1;
   6,  0, -4,  0,  3,  0, -2,  0,  1,  0, -1, 0, 1;
		

Links

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

Crossrefs

Cf. A104974, A104976 (row sums), A104977.

Programs

Mathematica

t[n_, k_]:= t[n, k]= If[k==n, 1, (1+(-1)^(n-k))/2 Sum[Binomial[k, j]*t[(n-k)/2, j],{j, (n-k)/2}]];
S[n_]:= Sum[(-1)^j*t[n, j], {j,0,n}]; (* S = A104977 *)
T[n_, k_]:= If[EvenQ[n-k], S[(n-k)/2], 0];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 08 2021 *)

Sage

@CachedFunction
def t(n,k): return 1 if (k==n) else ((1+(-1)^(n-k))/2)*sum( binomial(k, j)*t((n-k)/2, j) for j in (1..(n-k)//2) )
def S(n): return sum( (-1)^j*t(n, j) for j in (0..n) ) # S = A104977
def T(n,k): return S((n-k)/2) if (mod(n-k, 2)==0) else 0
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 08 2021

Formula

Riordan array (x^2/( (Sum_{k>=0} x^(2^k)) - x), x).

Sum_{k=0..n} T(n, k) = A104976(n).

T(n, k) = A104977((n-k)/2) if (n-k) is even, otherwise 0. - G. C. Greubel, Jun 08 2021