A257673 -id:A257673 - cp's OEIS frontend

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

Alois P. Heinz, Mar 11 2015

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].

			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, ...

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

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.

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);

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 *)

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).

A257674 INVERT transform of planar partitions.

Original entry on oeis.org

1, 1, 4, 13, 44, 144, 478, 1573, 5193, 17118, 56457, 186153, 613865, 2024192, 6674843, 22010313, 72579382, 239331323, 789198395, 2602391853, 8581422014, 28297352194, 93310894654, 307693910316, 1014624748161, 3345738548716, 11032617200372, 36380201398917

Offset: 0

Alois P. Heinz, May 03 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1930

Row sums of A257673.

Cf. A000219.

g:= proc(n) option remember; `if`(n=0, 1, add(
      g(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*g(i), i=1..n))
    end:
seq(a(n), n=0..36);

g[n_] := g[n] = If[n==0, 1, Sum[g[n-j] DivisorSigma[2, j], {j, 1, n}]/n];
a[n_] := a[n] = If[n==0, 1, Sum[a[n-i] g[i], {i, 1, n}]];
Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Aug 22 2021, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} A257673(n,k).
a(n) ~ c * d^n, where d = 3.2975132503126723336836261261699651439543806296893328114462016186843..., c = 0.3713883419445088444000361183895708557141471246022776707501762842135... . - Vaclav Kotesovec, May 19 2015
G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^k)^k). - Ilya Gutkovskiy, Oct 18 2018

cp's OEIS Frontend

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

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