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
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
Programs
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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 *)