0,3

Main diagonal (A121398) forms the partial sums of column 0 (A121399). The g.f. of the row sums is H(x)*(1-x)/(1-3x), where H(x) is the g.f. of column 0. This is the cascadence for function F(x) = 1 + x + x^2 when forced to form a triangle in which row n has n+1 terms for n>=0.

G.f.: A(x,y) = ( x*H(x) - y*H(x*y) )/( x*(1+y+y^2) - y ), where H(x) satisfies: H(x) = G*H(x*G)/x = g.f. of column 0 (A121399) and G/x is the g.f. of the Motzkin numbers (A001006): G = x*(1 + G + G^2).

Triangle begins:

1;

1, 2;

3, 3, 5;

6, 11, 8, 11;

17, 25, 30, 19, 28;

42, 72, 74, 77, 47, 70;

114, 188, 223, 198, 194, 117, 184;

302, 525, 609, 615, 509, 495, 301, 486;

827, 1436, 1749, 1733, 1619, 1305, 1282, 787, 1313;

2263, 4012, 4918, 5101, 4657, 4206, 3374, 3382, 2100, 3576;

6275, 11193, 14031, 14676, 13964, 12237, 10962, 8856, 9058, 5676, 9851;

The convolution of each row with [1,1,1] yields:

[1,1,1]*[1] = [1,1,1];

[1,1,1]*[1,2] = [1,3,3,2];

[1,1,1]*[3,3,5] = [3,6,11,8,5];

[1,1,1]*[6,11,8,11] = [6,17,25,30,19,11]; ...

Concatenate these convoluted rows after adding last and first terms:

1,1,1 + 1,3,3,2 + 3,6,11,8,5 + 6,17,25,30,19,11 + 17, ...

to obtain the concatenated rows of this original triangle:

1, 1,2, 3,3,5, 6,11,8,11, 17,25,30,19,28, ...

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

Cf. A121398 (main diagonal), A121399 (column 0), A001006 (Motzkin).

Sequence in context: A018051 A036803 A018131 this_sequence A056878 A092557 A081493

Adjacent sequences: A121397 A121398 A121399 this_sequence A121401 A121402 A121403

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 27 2006