A316101 Sequence a_k of column k shifts left when Weigh transform is applied k times with a_k(n) = n for n<2; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 3, 3, 1, 0, 1, 1, 1, 4, 6, 6, 1, 0, 1, 1, 1, 5, 10, 16, 12, 1, 0, 1, 1, 1, 6, 15, 32, 43, 25, 1, 0, 1, 1, 1, 7, 21, 55, 105, 120, 52, 1, 0, 1, 1, 1, 8, 28, 86, 210, 356, 339, 113, 1, 0, 1, 1, 1, 9, 36, 126, 371, 826, 1227, 985, 247, 1

Offset: 0

Author

Alois P. Heinz, Jun 24 2018

Keywords

• nonn
• tabl
• eigen

Examples

			Square array A(n,k) begins:
  0,  0,   0,   0,   0,    0,    0,    0,    0, ...
  1,  1,   1,   1,   1,    1,    1,    1,    1, ...
  1,  1,   1,   1,   1,    1,    1,    1,    1, ...
  1,  1,   1,   1,   1,    1,    1,    1,    1, ...
  1,  2,   3,   4,   5,    6,    7,    8,    9, ...
  1,  3,   6,  10,  15,   21,   28,   36,   45, ...
  1,  6,  16,  32,  55,   86,  126,  176,  237, ...
  1, 12,  43, 105, 210,  371,  602,  918, 1335, ...
  1, 25, 120, 356, 826, 1647, 2961, 4936, 7767, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, 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=0-10 give: A057427, A004111, A007561, A316103, A316104, A316105, A316106, A316107, A316108, A316109, A316110.

Rows include (offsets may differ): A000004, A000012, A000027, A000217, A134465.

Main diagonal gives A316102.

Cf. A144042, A316074.

Programs

Maple

wtr:= proc(p) local b; b:= proc(n, i) option remember;
       `if`(n=0, 1, `if`(i<1, 0, add(binomial(p(i), j)*
         b(n-i*j, i-1), j=0..n/i))) end: j-> b(j$2)
      end:
g:= proc(k) option remember; local b, t; b[0]:= j->
     `if`(j<2, j, b[k](j-1)); for t to k do
       b[t]:= wtr(b[t-1]) od: eval(b[0])
    end:
A:= (n, k)-> g(k)(n):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

wtr[p_] := Module[{b}, b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[p[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]]; b[#, #]&];
g[k_] := g[k] = Module[{b, t}, b[0][j_] := If[j < 2, j, b[k][j - 1]]; For[ t = 1, t <= k + 1, t++, b[t] = wtr[b[t - 1]]]; b[0]];
A[n_, k_] := g[k][n];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)