A255961 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j*k).

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 13, 0, 1, 5, 18, 37, 47, 24, 0, 1, 6, 25, 64, 111, 110, 48, 0, 1, 7, 33, 100, 215, 303, 258, 86, 0, 1, 8, 42, 146, 370, 660, 804, 568, 160, 0, 1, 9, 52, 203, 588, 1251, 1938, 2022, 1237, 282, 0

Offset

0,8

Comments

A(n,k) is the number of partitions of n when parts i are of k*i kinds. A(2,2) = 7: [2a], [2b], [2c], [2d], [1a,1a], [1a,1b], [1b,1b].

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

G.f. of column k: Product_{j>=1} 1/(1-x^j)^(j*k).

T(n,k) = Sum_{i=0..k} C(k,i) * A257673(n,k-i).

Example

Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,     1,     1, ...
  0,  1,   2,    3,    4,     5,     6,     7, ...
  0,  3,   7,   12,   18,    25,    33,    42, ...
  0,  6,  18,   37,   64,   100,   146,   203, ...
  0, 13,  47,  111,  215,   370,   588,   882, ...
  0, 24, 110,  303,  660,  1251,  2160,  3486, ...
  0, 48, 258,  804, 1938,  4005,  7459, 12880, ...
  0, 86, 568, 2022, 5400, 12150, 24354, 44885, ...

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 02 2016, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A000219, A161870, A255610, A255611, A255612, A255613, A255614, A193427, A316461, A316462.

Rows n=0-3 give: A000012, A001477, A055998, A101853.

Main diagonal gives A255672.

Antidiagonal sums give A299166.

Cf. A144064, A257673.

Sequence in context: A320782 A191588 A106450 * A297328 A378289 A362079

Adjacent sequences: A255958 A255959 A255960 * A255962 A255963 A255964

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 11 2015

Status

approved