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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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Alois P. Heinz, Rows 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 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 Columns k=1..10 give A000081, A007562, A218020, A218021, A218022, A218023, A218024, A218025, A218026, A218027. T(2n,n) gives A000041(n). Cf. A316074. Sequence in context: A278984 A111579 A144374 * A258709 A239144 A325528 Adjacent sequences:  A144015 A144016 A144017 * A144019 A144020 A144021 KEYWORD eigen,nonn,tabl AUTHOR Alois P. Heinz, Sep 07 2008 STATUS approved

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Last modified August 11 08:50 EDT 2020. Contains 336422 sequences. (Running on oeis4.)