A316074 Sequence a_k of column k shifts left k places under Weigh transform and equals signum(n) for n=1, 1<=k<=n, read by rows.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 6, 2, 2, 1, 1, 1, 12, 4, 2, 2, 1, 1, 1, 25, 6, 3, 2, 2, 1, 1, 1, 52, 10, 5, 3, 2, 2, 1, 1, 1, 113, 17, 7, 4, 3, 2, 2, 1, 1, 1, 247, 29, 10, 6, 4, 3, 2, 2, 1, 1, 1, 548, 51, 17, 8, 5, 4, 3, 2, 2, 1, 1, 1, 1226, 89, 26, 12, 7, 5, 4, 3, 2, 2, 1, 1, 1

Offset: 1

Author

Alois P. Heinz, Jun 23 2018

Keywords

• nonn
• tabl
• eigen

Examples

			Triangle T(n,k) begins:
    1;
    1,  1;
    1,  1, 1;
    2,  1, 1, 1;
    3,  2, 1, 1, 1;
    6,  2, 2, 1, 1, 1;
   12,  4, 2, 2, 1, 1, 1;
   25,  6, 3, 2, 2, 1, 1, 1;
   52, 10, 5, 3, 2, 2, 1, 1, 1;
  113, 17, 7, 4, 3, 2, 2, 1, 1, 1;
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Crossrefs

Columns k=1-10 give: A004111, A007560, A316075, A316076, A316077, A316078, A316079, A316080, A316081, A316082.

T(2n,n) gives A000009.

Cf. A057427, A144018, A316101.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(T(i, k), j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
T:= (n, k)-> `if`(n
				

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[T[i, k], j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
T[n_, k_] := If[n < k, Sign[n], b[n - k, n - k, k]];
Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)