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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.
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%I #30 Jun 23 2018 18:58:17

%S 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 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 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 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 Alois P. Heinz, <a href="/A144018/b144018.txt">Rows n = 1..141, flattened</a>

%H 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 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%e 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 Triangle begins:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 4, 2, 1, 1;

%e 9, 3, 2, 1, 1;

%e 20, 6, 3, 2, 1, 1;

%e 48, 10, 5, 3, 2, 1, 1;

%e 115, 20, 8, 5, 3, 2, 1, 1;

%e 286, 36, 14, 7, 5, 3, 2, 1, 1;

%e 719, 72, 23, 12, 7, 5, 3, 2, 1, 1;

%p etrk:= proc(p) proc(n, k) option remember; `if`(n=0, 1,

%p add(add(d*p(d, k), d=numtheory[divisors](j))*

%p procname(n-j, k), j=1..n)/n)

%p end end:

%p B:= etrk(T):

%p T:= (n, k)-> `if`(n<=k, `if`(n=0, 0, 1), B(n-k, k)):

%p seq(seq(T(n, k), k=1..n), n=1..14);

%t 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 Columns k=1..10 give A000081, A007562, A218020, A218021, A218022, A218023, A218024, A218025, A218026, A218027.

%Y T(2n,n) gives A000041(n).

%Y Cf. A316074.

%K eigen,nonn,tabl

%O 1,4

%A _Alois P. Heinz_, Sep 07 2008