A316103 -id:A316103 - cp's OEIS frontend

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

Alois P. Heinz, Jun 24 2018

			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, ...

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]

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.

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);

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 *)

A144035 Shifts left when Euler transform applied 3 times.

Original entry on oeis.org

0, 1, 1, 4, 13, 51, 197, 828, 3526, 15538, 69627, 317564, 1466868, 6853320, 32317354, 153636769, 735493374, 3542610091, 17155811156, 83480667707, 407969449010, 2001479974330, 9853652529076, 48666276924852, 241059431701113, 1197237945360797, 5960804381552300

Offset: 0

Alois P. Heinz, Sep 07 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

3rd column of A144042.

Cf. A316103.

b:= ((proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1,
        add(add(d*p(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
      end end)@@3)(a):
a:= n-> b(n-1):
seq(a(n), n=0..25);  # revised Alois P. Heinz, Aug 27 2018

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := Module[{a, b, t}, b[1] = etr[a]; For[ t = 2, t <= k, t++, b[t] = etr[b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]]; Table[ A[n, 3], {n, 0, 30} ] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

cp's OEIS Frontend

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.

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