A104975 - OEIS

%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