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%I #7 Sep 08 2022 08:45:30
%S 1,1,2,1,2,4,1,2,4,8,0,1,3,7,15,0,0,1,4,11,26,0,0,0,1,5,16,42,0,0,0,0,
%T 1,6,22,64,1,1,1,1,1,2,8,30,94,1,2,3,4,5,6,8,16,46,140,1,2,4,7,11,16,
%U 22,30,46,92,232,1,2,4,8,15,26,42,64,94,140,232,464,0,1,3,7,15,30,56,98
%N Triangular array T read by rows: T(j,1) = 1 for ((j-1) mod 8) < 4, else 0; T(j,k) = T(j-1,k-1) + T(j,k-1) for 2 <= k <= j.
%C All columns are periodic with period length 8. The (4+8*i)-th row equals the first (4+8*i) terms of the main diagonal (i >= 0). Main diagonal and eighth subdiagonal agree; generally j-th subdiagonal equals (j+8)-th subdiagonal.
%e First seven rows of T are
%e [ 1 ]
%e [ 1, 2 ]
%e [ 1, 2, 4 ]
%e [ 1, 2, 4, 8 ]
%e [ 0, 1, 3, 7, 15 ]
%e [ 0, 0, 1, 4, 11, 26 ]
%e [ 0, 0, 0, 1, 5, 16, 42 ].
%t T[j_, 1] := If[Mod[j-1, 8]<4, 1, 0]; T[j_, k_] := T[j, k] = T[j-1, k-1]+T[j, k-1]; Table[T[j, k], {j, 1, 13}, {k, 1, j}] // Flatten (* _Jean-François Alcover_, Mar 06 2014 *)
%o (PARI) {m=13; M=matrix(m, m); for(j=1, m, M[j, 1]=if((j-1)%8<4, 1, 0)); for(k=2, m, for(j=k, m, M[j, k]=M[j-1, k-1]+M[j, k-1])); for(j=1, m, for(k=1, j, print1(M[j, k], ",")))}
%o (Magma) m:=13; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 8 lt 4 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; &cat[ [ M[j, k]: k in [1..j] ]: j in [1..m] ];
%Y Cf. A131022, A129961 (main diagonal of T), A131075 (first subdiagonal of T), A131076 (row sums of T), A131077 (antidiagonal sums of T). First through sixth column of T are in A131078, A131079, A131080, A131081, A131082, A131083 resp.
%K nonn,tabl
%O 1,3
%A _Klaus Brockhaus_, following a suggestion of _Paul Curtz_, Jun 14 2007
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