0,4

Related to the number of subpartitions of a partition as defined in A115728; for examples involving column k of successive matrix powers, see A121430, A121431, A121432 and A121433. Essentially the same as triangle A101479, but this form best illustrates the nice properties of this triangle.

G.f.: Column k of successive powers of T satisfy the amazing relation given by: 1 = Sum_{n>=0}(1-x)^(n+1)*x^(n(n+1)/2+k*n)*Sum_{j=0..n+k}[T^(j+1)](n+k,k)*x^j.

Triangle T begins:

1;

1, 1;

3, 1, 1;

18, 4, 1, 1;

170, 30, 5, 1, 1;

2220, 335, 45, 6, 1, 1;

37149, 4984, 581, 63, 7, 1, 1;

758814, 92652, 9730, 924, 84, 8, 1, 1;

18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1;

508907970, 53636520, 4843125, 387567, 28365, 1965, 135, 10, 1, 1;

To get row 4 of T, append '1' to row 3 of matrix power T^5:

1;

5, 1;

25, 5, 1;

170, 30, 5, 1; ...

To get row 5 of T, append '1' to row 4 of matrix power T^6:

1;

6, 1;

33, 6, 1;

233, 39, 6, 1;

2220, 335, 45, 6, 1; ...

Likewise, get row n of T by appending '1' to row (n-1) of T^(n+1).

(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)[i-1, j]); )); A=B); return((A^1)[n+1, k+1])}

Cf. A121416 (T^2), A121420 (T^3), columns: A121413, A121414, A121415; related tables: A121424, A121426, A121428; related subpartitions: A121430, A121431, A121432, A121433.

Sequence in context: A060325 A135021 A087987 this_sequence A016561 A111382 A156950

Adjacent sequences: A121409 A121410 A121411 this_sequence A121413 A121414 A121415

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 30 2006