cp's OEIS Frontend

A316074 Sequence a_k of column k shifts left k places under Weigh transform and equals signum(n) for n triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A316074 #19 Jul 10 2018 08:16:16
%S A316074 1,1,1,1,1,1,2,1,1,1,3,2,1,1,1,6,2,2,1,1,1,12,4,2,2,1,1,1,25,6,3,2,2,
%T A316074 1,1,1,52,10,5,3,2,2,1,1,1,113,17,7,4,3,2,2,1,1,1,247,29,10,6,4,3,2,2,
%U A316074 1,1,1,548,51,17,8,5,4,3,2,2,1,1,1,1226,89,26,12,7,5,4,3,2,2,1,1,1
%N A316074 Sequence a_k of column k shifts left k places under Weigh transform and equals signum(n) for n<k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H A316074 Alois P. Heinz, <a href="/A316074/b316074.txt">Rows n = 1..200, flattened</a>
%H A316074 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 A316074 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]
%e A316074 Triangle T(n,k) begins:
%e A316074     1;
%e A316074     1,  1;
%e A316074     1,  1, 1;
%e A316074     2,  1, 1, 1;
%e A316074     3,  2, 1, 1, 1;
%e A316074     6,  2, 2, 1, 1, 1;
%e A316074    12,  4, 2, 2, 1, 1, 1;
%e A316074    25,  6, 3, 2, 2, 1, 1, 1;
%e A316074    52, 10, 5, 3, 2, 2, 1, 1, 1;
%e A316074   113, 17, 7, 4, 3, 2, 2, 1, 1, 1;
%p A316074 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A316074       add(binomial(T(i, k), j)*b(n-i*j, i-1, k), j=0..n/i)))
%p A316074     end:
%p A316074 T:= (n, k)-> `if`(n<k, signum(n), b(n-k$2, k)):
%p A316074 seq(seq(T(n, k), k=1..n), n=1..16);
%t A316074 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[T[i, k], j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];
%t A316074 T[n_, k_] := If[n < k, Sign[n], b[n - k, n - k, k]];
%t A316074 Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 10 2018, after _Alois P. Heinz_ *)
%Y A316074 Columns k=1-10 give: A004111, A007560, A316075, A316076, A316077, A316078, A316079, A316080, A316081, A316082.
%Y A316074 T(2n,n) gives A000009.
%Y A316074 Cf. A057427, A144018, A316101.
%K A316074 nonn,tabl,eigen
%O A316074 1,7
%A A316074 _Alois P. Heinz_, Jun 23 2018