A167271 - OEIS

%I #13 Sep 20 2025 16:03:28

%S 1,1,1,1,0,1,1,2,0,1,1,0,0,0,1,1,3,1,0,0,1,1,0,0,0,0,0,1,1,4,0,3,0,0,

%T 0,1,1,0,1,0,0,0,0,0,1,1,5,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,1,6,

%U 1,6,0,4,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1

%N Triangle read by rows: T(n,k) = A167269(n/k-1, k-1) if k divides n, T(n,k) = 0 otherwise.

%C To form column k+1, aerate the terms of column k of A167269 with k zeros.

%H Andrew Howroyd, <a href="/A167271/b167271.txt">Table of n, a(n) for n = 1..1275</a>

%e Triangle begins:

%e   1,

%e   1, 1;

%e   1, 0, 1;

%e   1, 2, 0,  1;

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

%e   1, 3, 1,  0, 0, 1;

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

%e   1, 4, 0,  3, 0, 0, 0, 1;

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

%e   1, 5, 0,  0, 1, 0, 0, 0, 0, 1;

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

%e   1, 6, 1,  6, 0, 4, 0, 0, 0, 0, 0, 1;

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

%e   1, 7, 0,  0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;

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

%e   1, 8, 0, 10, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1;

%e   ...

%o (PARI) T(n,k) = if(n%k==0, if(k%2, 1, binomial(n/k + k/2 - 1, k/2))) \\ _Andrew Howroyd_, Sep 20 2025

%Y Row sums are A167272.

%Y Infinite product of shifted columns gives A167273.

%Y Cf. A167269.

%K nonn,tabl

%O 1,8

%A _Gary W. Adamson_ & _Mats Granvik_, Oct 31 2009

%E Name edited and offset changed by _Andrew Howroyd_, Sep 20 2025