A132610 Triangle T, read by rows, where row n+1 of T = row n of matrix power T^(2n) with appended '1' for n>=0 with T(0,0)=1.

1, 1, 1, 2, 1, 1, 14, 4, 1, 1, 194, 39, 6, 1, 1, 4114, 648, 76, 8, 1, 1, 118042, 15465, 1510, 125, 10, 1, 1, 4274612, 483240, 41121, 2908, 186, 12, 1, 1, 186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1, 9577713250, 861282832, 59857416, 3437248, 171700, 7824, 344, 16, 1, 1

Offset

0,4

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

T(n+1,1) is divisible by n for n>=1.

Example

Triangle begins:
1;
1, 1;
2, 1, 1;
14, 4, 1, 1;
194, 39, 6, 1, 1;
4114, 648, 76, 8, 1, 1;
118042, 15465, 1510, 125, 10, 1, 1;
4274612, 483240, 41121, 2908, 186, 12, 1, 1;
186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...
GENERATE T FROM EVEN MATRIX POWERS OF T.
Matrix square T^2 begins:
1;
2, 1; <-- row 2 of T
5, 2, 1;
34, 9, 2, 1;
453, 88, 13, 2, 1; ...
where row 2 of T = row 1 of T^2 with appended '1'.
Matrix fourth powers T^4 begins:
1;
4, 1;
14, 4, 1; <-- row 3 of T
96, 22, 4, 1;
1215, 220, 30, 4, 1; ...
where row 3 of T = row 2 of T^4 with appended '1'.
Matrix sixth power T^6 begins:
1;
6, 1;
27, 6, 1;
194, 39, 6, 1; <-- row 4 of T
2394, 404, 51, 6, 1; ...
where row 4 of T = row 3 of T^6 with appended '1'.
ALTERNATE GENERATING METHOD.
Start with [1,0,0,0]; take partial sums and append 1 zero;
take partial sums twice more:
(1), 0, 0, 0;
1, 1, 1, (1), 0;
1, 2, 3, 4, (4);
1, 3, 6, 10, (14);
the final nonzero terms form row 3: [14,4,1,1].
Start with [1,0,0,0,0,0]; take partial sums and append 3 zeros;
take partial sums and append 1 zero; take partial sums twice more:
(1), 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, (1), 0, 0, 0;
1, 2, 3, 4, 5, 6, 6, 6, (6), 0;
1, 3, 6, 10, 15, 21, 27, 33, 39, (39);
1, 4, 10, 20, 35, 56, 83, 116, 155, (194);
the final nonzero terms form row 4: [194,39,6,1,1].
Continuing in this way produces all the rows of this triangle.

Maple

b:= proc(n) option remember;
      Matrix(n, (i, j)-> T(i-1, j-1))^(2*n-2)
    end:
T:= proc(n, k) option remember;
     `if`(n=k, 1, `if`(k>n, 0, b(n)[n, k+1]))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 13 2020

Mathematica

b[n_] := b[n] = MatrixPower[Table[T[i-1, j-1], {i, n}, {j, n}], 2n-2];
T[n_, k_] := T[n, k] = If[n == k, 1, If[k > n, 0, b[n][[n, k+1]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)

Prog

(PARI) {T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k-1, p=(n-1)^2-(n-j-1)^2; A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A=Vec((Polrev(A)+x*O(x^p))/(1-x)); A[p+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. columns: A132611, A132612, A132613; A132614; variants: A132615, A101479.

Cf. A304189, A304192, A304188.

Sequence in context: A176291 A337514 A054505 * A132625 A164792 A132318

Adjacent sequences: A132607 A132608 A132609 * A132611 A132612 A132613

Keyword

nonn,tabl

Author

Paul D. Hanna, Aug 23 2007

Status

approved