0,8

See definition of pendular triangle and pendular sums at A118340.

T(2*n+m,n) = [A118351^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of the central terms A118351.

Row 6 equals the pendular sums of row 5,

[1, 4,13, 7, 1, 0], where the sums proceed as follows:

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

[1,__,__,__,__, 1]: T(6,5) = T(6,0) + 3*T(5,5) = 1 + 3*0 = 1;

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

[1, 5,__,__, 8, 1]: T(6,4) = T(6,1) + 3*T(5,4) = 5 + 3*1 = 8;

[1, 5,21,__, 8, 1]: T(6,2) = T(6,4) + T(5,2) = 8 + 13 = 21;

[1, 5,21,42, 8, 1]: T(6,3) = T(6,2) + 3*T(5,3) = 21 + 3*7 = 42;

[1, 5,21,42, 8, 1, 0] finally, append a zero to obtain row 6.

Triangle begins:

1;

1, 0;

1, 1, 0;

1, 2, 1, 0;

1, 3, 6, 1, 0;

1, 4, 13, 7, 1, 0;

1, 5, 21, 42, 8, 1, 0;

1, 6, 30, 96, 54, 9, 1, 0;

1, 7, 40, 163, 325, 67, 10, 1, 0;

1, 8, 51, 244, 770, 445, 81, 11, 1, 0;

1, 9, 63, 340, 1353, 2688, 583, 96, 12, 1, 0;

1, 10, 76, 452, 2093, 6530, 3842, 740, 112, 13, 1, 0;

1, 11, 90, 581, 3010, 11760, 23286, 5230, 917, 129, 14, 1, 0; ...

Central terms are T(2*n,n) = A118351(n);

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

T(2*n+1,n) = A118352(n) = [A118351^2](n),

T(2*n+2,n) = A118353(n) = [A118351^3](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)+3*T(n-1, k)))))}

Cf. A118351, A118352, A118353, A118354.

Sequence in context: A071921 A003992 A118345 this_sequence A109970 A116088 A136501

Adjacent sequences: A118347 A118348 A118349 this_sequence A118351 A118352 A118353

nonn,tabl

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