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A144374 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k begins with (k+1) 1's and a_k(n) shifts k places down under Dirichlet convolution. 9

%I

%S 1,1,1,2,1,1,4,2,1,1,9,2,2,1,1,18,5,2,2,1,1,40,4,3,2,2,1,1,80,12,4,3,

%T 2,2,1,1,168,8,6,2,3,2,2,1,1,340,28,6,6,2,3,2,2,1,1,698,17,10,4,4,2,3,

%U 2,2,1,1,1396,60,13,8,4,4,2,3,2,2,1,1,2844,34,16,5,6,2,4,2,3,2,2,1,1,5688

%N Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k begins with (k+1) 1's and a_k(n) shifts k places down under Dirichlet convolution.

%C Sequence a_k of column k begins with k terms from A000012 (only the last is in the triangle), followed by the first (k+1) terms from A000005.

%H Alois P. Heinz, <a href="/A144374/b144374.txt">Rows n = 1..141, flattened</a>

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

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 4, 2, 1, 1;

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

%e 18, 5, 2, 2, 1, 1;

%p with(numtheory): dck:= proc(b,c) proc(n, k) option remember; add(b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: seq(seq(T(n, k), k=1..n), n=1..14);

%t dck[b_, c_][n_, k_] := dck[b, c][n, k] = Sum[b[d, k]*c[n/d, k], {d, If[n < 0, {}, Divisors[n]]}]; B = dck[T, T]; T[n_, k_] := If[n <= k, 1, B[n-k, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* _Jean-Fran├žois Alcover_, Jan 15 2014, translated from Maple *)

%Y Columns 1-9 give: A038044, A144366, A144367, A144368, A144369, A144370, A144371, A144372, A144373.

%Y Cf. A000012, A000005.

%K eigen,nonn,tabl

%O 1,4

%A _Alois P. Heinz_, Sep 18 2008

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Last modified April 13 18:29 EDT 2021. Contains 342937 sequences. (Running on oeis4.)