A218027 -id:A218027 - cp's OEIS frontend

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.

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

Alois P. Heinz, Sep 07 2008

			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;

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

Columns k=1..10 give A000081, A007562, A218020, A218021, A218022, A218023, A218024, A218025, A218026, A218027.
T(2n,n) gives A000041(n).
Cf. A316074.

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);

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 *)

A316082 Sequence shifts left ten places under Weigh transform with a(n) = signum(n) for n<10.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 19, 24, 31, 40, 51, 67, 87, 114, 149, 195, 255, 334, 438, 574, 753, 989, 1300, 1711, 2255, 2973, 3927, 5190, 6867, 9092, 12046, 15971, 21188, 28124, 37351, 49631, 65981, 87763, 116794

Offset: 0

Alois P. Heinz, Jun 23 2018

Alois P. Heinz, Table of n, a(n) for n = 0..2500

Column k=10 of A316074.

Cf. A057427, A218027.

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

G.f.: x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 * Product_{n>=1} (1 + x^n)^a(n). - Ilya Gutkovskiy, May 09 2019

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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.

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