cp's OEIS Frontend

A104975 Inverse of a Fredholm-Rueppel triangle.

%I A104975 #8 Jun 09 2021 02:28:02
%S A104975 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,
%T A104975 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,
%U A104975 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
%N A104975 Inverse of a Fredholm-Rueppel triangle.
%C A104975 Sequence array for A104977.
%C A104975 Inverse of A104974.
%H A104975 G. C. Greubel, <a href="/A104975/b104975.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A104975 Riordan array (x^2/( (Sum_{k>=0} x^(2^k)) - x), x).
%F A104975 Sum_{k=0..n} T(n, k) = A104976(n).
%F A104975 T(n, k) = A104977((n-k)/2) if (n-k) is even, otherwise 0. - _G. C. Greubel_, Jun 08 2021
%e A104975 Triangle begins as:
%e A104975    1;
%e A104975    0,  1;
%e A104975   -1,  0,  1;
%e A104975    0, -1,  0,  1;
%e A104975    1,  0, -1,  0,  1;
%e A104975    0,  1,  0, -1,  0,  1;
%e A104975   -2,  0,  1,  0, -1,  0,  1;
%e A104975    0, -2,  0,  1,  0, -1,  0,  1;
%e A104975    3,  0, -2,  0,  1,  0, -1,  0,  1;
%e A104975    0,  3,  0, -2,  0,  1,  0, -1,  0,  1;
%e A104975   -4,  0,  3,  0, -2,  0,  1,  0, -1,  0,  1;
%e A104975    0, -4,  0,  3,  0, -2,  0,  1,  0, -1,  0, 1;
%e A104975    6,  0, -4,  0,  3,  0, -2,  0,  1,  0, -1, 0, 1;
%t A104975 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}]];
%t A104975 S[n_]:= Sum[(-1)^j*t[n, j], {j,0,n}]; (* S = A104977 *)
%t A104975 T[n_, k_]:= If[EvenQ[n-k], S[(n-k)/2], 0];
%t A104975 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 08 2021 *)
%o A104975 (Sage)
%o A104975 @CachedFunction
%o A104975 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) )
%o A104975 def S(n): return sum( (-1)^j*t(n, j) for j in (0..n) ) # S = A104977
%o A104975 def T(n,k): return S((n-k)/2) if (mod(n-k, 2)==0) else 0
%o A104975 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 08 2021
%Y A104975 Cf. A104974, A104976 (row sums), A104977.
%K A104975 easy,sign,tabl
%O A104975 0,22
%A A104975 _Paul Barry_, Mar 30 2005