0,7

Remarkably, T equals the product of these triangular matrices: T = A107867*A107862^-1 = A107870*A107867^-1 = A107873*A107870^-1; reversing the order of these products produces triangle A107876.

Rows begin:

[1],

[1,1],

[1,1,1],

[3,2,1,1],

[19,9,3,1,1],

[191,70,18,4,1,1],

[2646,795,170,30,5,1,1],

[46737,11961,2220,335,45,6,1,1],

[1003150,224504,37149,4984,581,63,7,1,1],...

Row 4 starts with row 3 of T^3 which begins:

[1],

[3,1],

[6,3,1],

[19,9,3,1],...

row 5 starts with row 4 of T^4 which begins:

[1],

[4,1],

[10,4,1],

[34,14,4,1],

[191,70,18,4,1],...

Appending a '1' to each forms the diagonal of T.

An alternate generating method is illustrated as follows.

For row 4:

Start with a '1' and append 2 zeros,

take partial sums and append 1 zero,

take partial sums thrice more, resulting in:

1, 0, 0;

1, 1, 1, 0;

1, 2, 3, 3;

1, 3, 6, 9;

1, 4,10,19.

Final non-zero terms form row 4: [19,9,3,1,1].

For row 5:

Start with a '1' and append 3 zeros,

take partial sums and append 2 zeros,

take partial sums and append 1 zero,

take partial sums thrice more, resulting in:

1, 0, 0, 0;

1, 1, 1, 1, 0,` 0;

1, 2, 3, 4, 4,` 4,` 0;

1, 3, 6,10,14, 18, 18;

1, 4,10,20,34, 52, 70;

1, 5,15,35,69,121,191.

Final non-zero terms form row 5: [191,70,18,4,1,1].

Likewise, for row 6:

1, 0, 0, 0,` 0;

1, 1, 1, 1,` 1,` 0,` 0,` 0;

1, 2, 3, 4,` 5,` 5,` 5,` 5,`` 0,`` 0;

1, 3, 6,10, 15, 20, 25, 30,` 30,` 30,`` 0;

1, 4,10,20, 35, 55, 80,110, 140, 170, 170;

1, 5,15,35, 70,125,205,315, 455, 625, 795;

1, 6,21,56,126,251,456,771,1226,1851,2646.

Final non-zero terms form row 6: [2646,795,170,30,5,1,1].

Continuing in this way generates all rows of this triangle.

(PARI) {T(n, k)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(A^(i-2))[i-1, j]); )); A=B); return(A[n+1, k+1])}

(PARI) {T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k-1, p=(n-1)*(n-2)/2-(n-j-1)*(n-j-2)/2; A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A=Vec((Polrev(A)+x*O(x^p))/(1-x)); A[p+1]}

Columns are A101481, A101482, A101483, row sums form A101484.

Cf. A107876 (dual triangle).

Sequence in context: A111760 A078424 A092742 this_sequence A136170 A101221 A137241

Adjacent sequences: A101476 A101477 A101478 this_sequence A101480 A101481 A101482

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 21 2005, Jul 26 2006, May 27 2007