%I #18 May 08 2019 00:20:57

%S 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,

%T 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,

%U 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

%N 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.

%H Alois P. Heinz, <a href="/A316101/b316101.txt">Antidiagonals n = 0..140, flattened</a>

%H M. Bernstein and N. J. A. Sloane, <a href="https://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%e Square array A(n,k) begins:

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

%e 1, 3, 6, 10, 15, 21, 28, 36, 45, ...

%e 1, 6, 16, 32, 55, 86, 126, 176, 237, ...

%e 1, 12, 43, 105, 210, 371, 602, 918, 1335, ...

%e 1, 25, 120, 356, 826, 1647, 2961, 4936, 7767, ...

%p wtr:= proc(p) local b; b:= proc(n, i) option remember;

%p `if`(n=0, 1, `if`(i<1, 0, add(binomial(p(i), j)*

%p b(n-i*j, i-1), j=0..n/i))) end: j-> b(j$2)

%p end:

%p g:= proc(k) option remember; local b, t; b[0]:= j->

%p `if`(j<2, j, b[k](j-1)); for t to k do

%p b[t]:= wtr(b[t-1]) od: eval(b[0])

%p end:

%p A:= (n, k)-> g(k)(n):

%p seq(seq(A(n, d-n), n=0..d), d=0..14);

%t 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[#, #]&];

%t 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]];

%t A[n_, k_] := g[k][n];

%t Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Jul 10 2018, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A057427, A004111, A007561, A316103, A316104, A316105, A316106, A316107, A316108, A316109, A316110.

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

%Y Main diagonal gives A316102.

%Y Cf. A144042, A316074.

%K nonn,tabl,eigen

%O 0,20

%A _Alois P. Heinz_, Jun 24 2018