A104974 - OEIS

%I #17 Sep 08 2022 08:45:17

%S 1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0,0,1,0,1,1,0,0,0,1,0,1,0,1,0,0,0,1,

%T 0,1,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,1,0,1,0,0,0,0,1,0,0,0,1,0,1,0,0,

%U 0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,1

%N A Fredholm-Rueppel triangle.

%C Sequence matrix for A036987(n+1).

%C Riordan array ( (Sum_{k>=0} x^(2^k)/x^2) - 1/x, x).

%C Diagonal sums are A070939(n+1), with interpolated zeros.

%C Inverse is A104975.

%H G. C. Greubel, <a href="/A104974/b104974.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = A000108(n+1-k) mod 2. [Corrected by _R. J. Mathar_, Apr 21 2021]

%F Sum_{k=0..n} T(n, k) = A000523(n+1).

%e Triangle begins as:

%e   1;

%e   0, 1;

%e   1, 0, 1;

%e   0, 1, 0, 1;

%e   0, 0, 1, 0, 1;

%e   0, 0, 0, 1, 0, 1;

%e   1, 0, 0, 0, 1, 0, 1;

%e   0, 1, 0, 0, 0, 1, 0, 1;

%e   0, 0, 1, 0, 0, 0, 1, 0, 1;

%e   0, 0, 0, 1, 0, 0, 0, 1, 0, 1;

%e   0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1;

%e   0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1;

%e   0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1;

%e   0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1;

%p A104974 := proc(n,k)

%p     modp(A000108(n+1-k),2);

%p end proc:

%p seq(seq( A104974(n,k), k=0..n), n=0..15); # _R. J. Mathar_, Apr 21 2021

%t Table[Mod[CatalanNumber[n-k+1], 2], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 08 2021 *)

%o (Magma) [(Catalan(n-k+1) mod 2): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jun 08 2021

%o (Sage) flatten([[mod(catalan_number(n-k+1), 2) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jun 08 2021

%Y Cf. A000523 (row sums), A036987, A070939, A104975.

%K easy,nonn,tabl

%O 0,1

%A _Paul Barry_, Mar 30 2005