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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.
11
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, 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, 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
OFFSET
1,4
LINKS
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
EXAMPLE
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.
Triangle begins:
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, 5, 3, 2, 1, 1;
286, 36, 14, 7, 5, 3, 2, 1, 1;
719, 72, 23, 12, 7, 5, 3, 2, 1, 1;
MAPLE
etrk:= proc(p) proc(n, k) option remember; `if`(n=0, 1,
add(add(d*p(d, k), d=numtheory[divisors](j))*
procname(n-j, k), j=1..n)/n)
end end:
B:= etrk(T):
T:= (n, k)-> `if`(n<=k, `if`(n=0, 0, 1), B(n-k, k)):
seq(seq(T(n, k), k=1..n), n=1..14);
MATHEMATICA
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 *)
CROSSREFS
T(2n,n) gives A000041(n).
Cf. A316074.
Sequence in context: A278984 A111579 A144374 * A258709 A239144 A325528
KEYWORD
eigen,nonn,tabl
AUTHOR
Alois P. Heinz, Sep 07 2008
STATUS
approved