A131074 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.

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, 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, 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

Offset: 1

Author

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 14 2007

Keywords

• nonn
• tabl

Comments

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.

Examples

			First seven rows of T are
[ 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 ].
		

Crossrefs

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.

Programs

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] ];

Mathematica

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 *)

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], ",")))}