A144042 Square array A(n,k), n>=1, k>=1, read by antidiagonals, with A(1,k)=1 and sequence a_k of column k shifts left when Euler transform applied k times.

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 9, 1, 1, 5, 13, 25, 20, 1, 1, 6, 19, 51, 77, 48, 1, 1, 7, 26, 89, 197, 258, 115, 1, 1, 8, 34, 141, 410, 828, 871, 286, 1, 1, 9, 43, 209, 751, 2052, 3526, 3049, 719, 1, 1, 10, 53, 295, 1260, 4337, 10440, 15538, 10834, 1842, 1, 1, 11, 64

Offset: 1

Author

Alois P. Heinz, Sep 08 2008

Keywords

• eigen
• nonn
• tabl

Examples

			Square array begins:
    1,   1,    1,     1,     1,     1,      1,      1, ...
    1,   1,    1,     1,     1,     1,      1,      1, ...
    2,   3,    4,     5,     6,     7,      8,      9, ...
    4,   8,   13,    19,    26,    34,     43,     53, ...
    9,  25,   51,    89,   141,   209,    295,    401, ...
   20,  77,  197,   410,   751,  1260,   1982,   2967, ...
   48, 258,  828,  2052,  4337,  8219,  14379,  23659, ...
  115, 871, 3526, 10440, 25512, 54677, 106464, 192615, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..141, 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]
• N. J. A. Sloane, Transforms

Crossrefs

Columns k=1-10 give: A000081, A007563, A144035, A144036, A144037, A144038, A144039, A144040, A144041, A305727.

Rows n=2-4 give: A000012, A000027, A034856.

Main diagonal gives A305725.

Cf. A316101.

Programs

Maple

etr:= 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:
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]:= etr(b[t-1]) od: eval(b[0])
    end:
A:= (n, k)-> g(k)(n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # revised Alois P. Heinz, Aug 27 2018

Mathematica

etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, 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[Table[A[n, 1 + d-n], {n, 1, d}], {d, 1, 14}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)