0,8

See A118340 for definition of pendular triangles and pendular sums.

T(2*n+m,n) = [A118346^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A118346.

Row 6 equals the pendular sums of row 5:

[1, 4,11, 6, 1, 0], where the pendular sums proceed as follows:

[1,__,__,__,__,__]: T(6,0) = T(5,0) = 1;

[1,__,__,__,__, 1]: T(6,5) = T(6,0) + 2*T(5,5) = 1 + 2*0 = 1;

[1, 5,__,__,__, 1]: T(6,1) = T(6,5) + T(5,1) = 1 + 4 = 5;

[1, 5,__,__, 7, 1]: T(6,4) = T(6,1) + 2*T(5,4) = 5 + 2*1 = 7;

[1, 5,18,__, 7, 1]: T(6,2) = T(6,4) + T(5,2) = 7 + 11 = 18;

[1, 5,18,30, 7, 1]: T(6,3) = T(6,2) + 2*T(5,3) = 18 + 2*6 = 30;

[1, 5,18,30, 7, 1, 0] finally, append a zero to obtain row 6.

Triangle begins:

1;

1, 0;

1, 1, 0;

1, 2, 1, 0;

1, 3, 5, 1, 0;

1, 4, 11, 6, 1, 0;

1, 5, 18, 30, 7, 1, 0;

1, 6, 26, 70, 40, 8, 1, 0;

1, 7, 35, 121, 201, 51, 9, 1, 0;

1, 8, 45, 184, 487, 286, 63, 10, 1, 0;

1, 9, 56, 260, 873, 1445, 386, 76, 11, 1, 0;

1, 10, 68, 350, 1375, 3592, 2147, 502, 90, 12, 1, 0; ...

Central terms are T(2*n,n) = A118346(n);

semi-diagonals form successive self-convolutions of the central terms:

T(2*n+1,n) = A118347(n) = [A118346^2](n),

T(2*n+2,n) = A118348(n) = [A118346^3](n).

(PARI) {T(n, k)=if(n<k|k<0, 0, if(k==0, 1, if(n==k, 0, if(n>2*k, T(n, n-k)+T(n-1, k), T(n, n-1-k)+2*T(n-1, k)))))}

Cf. A118346, A118347, A118348, A118349, A118340.

Sequence in context: A118340 A071921 A003992 this_sequence A118350 A109970 A116088

Adjacent sequences: A118342 A118343 A118344 this_sequence A118346 A118347 A118348

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 26 2006