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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,20
LINKS
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
Rows include (offsets may differ): A000004, A000012, A000027, A000217, A134465.
Main diagonal gives A316102.
Sequence in context: A219967 A060505 A336727 * A211452 A035188 A342148
KEYWORD
nonn,tabl,eigen
AUTHOR
Alois P. Heinz, Jun 24 2018
STATUS
approved

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Last modified May 20 02:28 EDT 2024. Contains 372703 sequences. (Running on oeis4.)