A316081 -id:A316081 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

Alois P. Heinz, Jun 23 2018

			Triangle T(n,k) begins:
    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, 1, 1, 1;
   52, 10, 5, 3, 2, 2, 1, 1, 1;
  113, 17, 7, 4, 3, 2, 2, 1, 1, 1;

Alois P. Heinz, Rows n = 1..200, 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]

Columns k=1-10 give: A004111, A007560, A316075, A316076, A316077, A316078, A316079, A316080, A316081, A316082.
T(2n,n) gives A000009.
Cf. A057427, A144018, A316101.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(T(i, k), j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
T:= (n, k)-> `if`(n

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[n_, k_] := If[n < k, Sign[n], b[n - k, n - k, k]];
Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2018, after Alois P. Heinz *)

A218026 Shifts 9 places left under Euler transform with a(0)=0 and a(n)=1 for n<9.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 57, 80, 109, 152, 211, 296, 415, 588, 832, 1185, 1689, 2413, 3449, 4940, 7073, 10141, 14544, 20880, 29991, 43131, 62064, 89417, 128925, 186090, 268808, 388677, 562381, 814393, 1180070, 1711131

Offset: 0

Alois P. Heinz, Oct 18 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Column k=9 of A144018.

Cf. A316081.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1,
      (add(add(d*a(d), d=divisors(j)) *b(n-j), j=1..n))/n)
    end:
a:= n-> `if`(n<9, signum(n), b(n-9)):
seq(a(n), n=0..60);

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 / Product_{n>=1} (1 - x^n)^a(n). - Ilya Gutkovskiy, May 08 2019

cp's OEIS Frontend

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

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