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A144018 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where sequence a_k of column k has a_k(0)=0, followed by (k+1)-fold 1 and a_k(n) shifts k places left under Euler transform.

%I A144018 #30 Jun 23 2018 18:58:17
%S A144018 1,1,1,2,1,1,4,2,1,1,9,3,2,1,1,20,6,3,2,1,1,48,10,5,3,2,1,1,115,20,8,
%T A144018 5,3,2,1,1,286,36,14,7,5,3,2,1,1,719,72,23,12,7,5,3,2,1,1,1842,137,40,
%U A144018 18,11,7,5,3,2,1,1,4766,275,69,30,16,11,7,5,3,2,1,1,12486,541,121,47,25,15,11,7,5,3,2,1,1
%N A144018 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where sequence a_k of column k has a_k(0)=0, followed by (k+1)-fold 1 and a_k(n) shifts k places left under Euler transform.
%H A144018 Alois P. Heinz, <a href="/A144018/b144018.txt">Rows n = 1..141, flattened</a>
%H A144018 M. Bernstein and N. J. A. Sloane, <a href="https://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H A144018 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A144018 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%e A144018 T(5,1) = ([1,2,4]*[1,1,4] + [1]*[1]*4 + [1,2]*[1,1]*2 + [1,3]*[1,2]*1)/4 = 36/4 = 9.
%e A144018 Triangle begins:
%e A144018     1;
%e A144018     1,  1;
%e A144018     2,  1,  1;
%e A144018     4,  2,  1,  1;
%e A144018     9,  3,  2,  1, 1;
%e A144018    20,  6,  3,  2, 1, 1;
%e A144018    48, 10,  5,  3, 2, 1, 1;
%e A144018   115, 20,  8,  5, 3, 2, 1, 1;
%e A144018   286, 36, 14,  7, 5, 3, 2, 1, 1;
%e A144018   719, 72, 23, 12, 7, 5, 3, 2, 1, 1;
%p A144018 etrk:= proc(p) proc(n, k) option remember; `if`(n=0, 1,
%p A144018          add(add(d*p(d, k), d=numtheory[divisors](j))*
%p A144018          procname(n-j, k), j=1..n)/n)
%p A144018        end end:
%p A144018 B:= etrk(T):
%p A144018 T:= (n, k)-> `if`(n<=k, `if`(n=0, 0, 1), B(n-k, k)):
%p A144018 seq(seq(T(n, k), k=1..n), n=1..14);
%t A144018 etrk[p_] := Module[{f}, f[n_, k_] := f[n, k] = If[n == 0, 1, (Sum[Sum[d*p[d, k], {d, Divisors[j]}]*f[n-j, k], {j, 1, n-1}] + Sum[d*p[d, k], {d, Divisors[n]}])/n]; f]; b = etrk[t]; t[n_, k_] := If[n <= k, If[n == 0, 0, 1], b[n-k, k]]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 01 2013, after _Alois P. Heinz_ *)
%Y A144018 Columns k=1..10 give A000081, A007562, A218020, A218021, A218022, A218023, A218024, A218025, A218026, A218027.
%Y A144018 T(2n,n) gives A000041(n).
%Y A144018 Cf. A316074.
%K A144018 eigen,nonn,tabl
%O A144018 1,4
%A A144018 _Alois P. Heinz_, Sep 07 2008