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

0,20

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]

Example

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, ...

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 *)

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.

Sequence in context: A219967 A060505 A336727 * A211452 A035188 A342148

Adjacent sequences: A316098 A316099 A316100 * A316102 A316103 A316104

Keyword

nonn,tabl,eigen

Author

Alois P. Heinz, Jun 24 2018

Status

approved