1,6

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

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, ...

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

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 *)

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.

Sequence in context: A387299 A096465 A124460 * A122084 A104559 A080853

Adjacent sequences: A144039 A144040 A144041 * A144043 A144044 A144045

eigen,nonn,tabl

Alois P. Heinz, Sep 08 2008

approved