A121412 Triangular matrix T, read by rows, where row n of T equals row (n-1) of T^(n+1) with an appended '1'.

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

Offset

0,4

Comments

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.

Links

Alois P. Heinz, Rows n = 0..45, flattened

Formula

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.

Example

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).

Mathematica

T[n_, k_] := Module[{A = {{1}}, B}, Do[B = Array[0&, {m, m}]; Do[Do[B[[i, j]] = If[j == i, 1, MatrixPower[A, i][[i-1, j]]], {j, 1, i}], {i, 1, m}]; A = B, {m, 1, n+1}]; A[[n+1, k+1]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 03 2019 *)

Prog

(PARI) {T(n, k) = my(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])}
for(n=0, 12, for(k=0, n, print1( T(n, k), ", ")); print(""))

Crossrefs

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: A333560 A176293 A176339 * A212855 A016561 A111382

Adjacent sequences: A121409 A121410 A121411 * A121413 A121414 A121415

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 30 2006

Status

approved