A066448 - OEIS

%I #28 Jan 13 2022 09:54:16

%S 1,0,1,0,2,0,0,2,0,0,0,2,1,0,0,0,2,2,0,0,0,0,2,4,0,0,0,0,0,2,6,0,0,0,

%T 0,0,0,2,8,0,0,0,0,0,0,0,2,10,1,0,0,0,0,0,0,0,2,12,2,0,0,0,0,0,0,0,0,

%U 2,14,4,0,0,0,0,0,0,0,0,0,2,16,8,0,0,0,0,0,0,0,0,0

%N Triangle T(n,k) giving number of basis partitions of n with a Durfee square of order k (n >= 0, 0 <= k <= n).

%H Seiichi Manyama, <a href="/A066448/b066448.txt">Rows n = 0..139, flattened</a>

%H J. M. Nolan, C. D. Savage and H. S. Wilf, <a href="https://doi.org/10.1016/s0012-365x(97)00101-5">Basis partitions</a>, Discrete Math. 179 (1998), 277-283.

%e Triangle begins:

%e   1;

%e   0, 1;

%e   0, 2, 0;

%e   0, 2, 0, 0;

%e   0, 2, 1, 0, 0;

%e   0, 2, 2, 0, 0, 0;

%e   0, 2, 4, 0, 0, 0, 0;

%e   0, 2, 6, 0, 0, 0, 0, 0;

%e   0, 2, 8, 0, 0, 0, 0, 0, 0;

%e   ...

%p T := proc(n,d); option remember; if n=0 and d=0 then RETURN(1) elif n<=0 or d<=0 then RETURN(0) else RETURN(T(n-d,d)+T(n-2*d+1,d-1)+T(n-3*d+1,d-1)) fi:

%o (PARI) T(n,k)=if(k<0||k>n,0,if(k==0,n==0,T(n-k,k)+T(n-2*k+1,k-1)+T(n-3*k+1,k-1))) /* _Michael Somos_, Mar 10 2004 */

%Y Row sums give A066447.

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_, Dec 29 2001