0,11

The diagonals may be generated by iterated convolutions of a base sequence B with the sequence C of central terms. The g.f. B(x) of the base sequence satisfies: B = 1 + x*B^2 + x^2*(B^2 - B); the g.f. C(x) of the central terms satisfies: C(x) = 1/(1+x - x*B(x)).

To obtain row 4, pendular sums of row 3 are carried out as follows.

[1,2,3,2,1,0,0]: given row 3;

[1,_,_,_,_,_,_]: start with T(4,0) = T(3,0) = 1;

[1,_,_,_,_,_,1]: T(4,6) = T(4,0) + T(3,6) = 1 + 0 = 1;

[1,3,_,_,_,_,1]: T(4,1) = T(4,6) + T(3,1) = 1 + 2 = 3;

[1,3,_,_,_,3,1]: T(4,5) = T(4,1) + T(3,5) = 3 + 0 = 3;

[1,3,6,_,_,3,1]: T(4,2) = T(4,5) + T(3,2) = 3 + 3 = 6;

[1,3,6,_,7,3,1]: T(4,4) = T(4,2) + T(3,4) = 6 + 1 = 7;

[1,3,6,9,7,3,1]: T(4,3) = T(4,4) + T(3,3) = 7 + 2 = 9;

[1,3,6,9,7,3,1,0,0]: complete row 4 by appending two zeros.

Triangle begins:

1;

1, 0, 0;

1, 1, 1, 0, 0;

1, 2, 3, 2, 1, 0, 0;

1, 3, 6, 9, 7, 3, 1, 0, 0;

1, 4, 10, 20, 30, 23, 11, 4, 1, 0, 0;

1, 5, 15, 36, 70, 104, 81, 40, 16, 5, 1, 0, 0;

1, 6, 21, 58, 133, 253, 374, 293, 149, 63, 22, 6, 1, 0, 0;

1, 7, 28, 87, 226, 501, 938, 1380, 1087, 564, 248, 93, 29, 7, 1, 0,0;

Central terms are:

C = A119371 = [1, 0, 1, 2, 7, 23, 81, 293, 1087, 4110, ...].

Lower diagonals start:

D1 = A119372 = [1, 1, 3, 9, 30, 104, 374, 1380, 5197, ...];

D2 = A119373 = [1, 2, 6, 20, 70, 253, 938, 3546, 13617, ...].

Diagonals above central terms (ignoring leading zeros) start:

U1 = A119375 = [1, 3, 11, 40, 149, 564, 2166, 8420, ...];

U2 = A119376 = [1, 4, 16, 63, 248, 980, 3894, 15563, ...].

There exists the base sequence:

B = A119370 = [1, 1, 2, 6, 19, 64, 225, 816, 3031, 11473, ...]

which generates all diagonals by convolutions with central terms:

D2 = B * D1 = B^2 * C

U2 = B * U1 = B^2 * C"

where C" = [1, 2, 7, 23, 81, 293, 1087, ...]

are central terms not including the initial [1,0].

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

Cf. A119370, A119371, A119372, A119373, A119374, A119375, A119376; variants: A118340, A118345, A118350.

Sequence in context: A017868 A017858 A167769 this_sequence A122445 A165592 A059285

Adjacent sequences: A119366 A119367 A119368 this_sequence A119370 A119371 A119372

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), May 16 2006