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