A091359 -id:A091359 - cp's OEIS frontend

A091355 Triangle read by rows: T(n,k) = number of planar partitions of n with k rows.

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 7, 9, 5, 2, 1, 11, 18, 11, 5, 2, 1, 15, 30, 22, 11, 5, 2, 1, 22, 53, 42, 24, 11, 5, 2, 1, 30, 85, 78, 46, 24, 11, 5, 2, 1, 42, 139, 138, 90, 48, 24, 11, 5, 2, 1, 56, 215, 239, 164, 94, 48, 24, 11, 5, 2, 1, 77, 336, 405, 298, 176, 96, 48, 24, 11, 5, 2, 1, 101, 504, 669, 520, 324, 180, 96, 48, 24, 11, 5, 2, 1

Offset: 1

Christian G. Bower, Jan 02 2004

Row sums give A000219.
Columns 1-5 are respectively A000041, A091356, A091357, A091358, and A091359.
Columns converge to A091360.

			Triangle starts:
01:  1,
02:  2, 1,
03:  3, 2, 1,
04:  5, 5, 2, 1,
05:  7, 9, 5, 2, 1,
06:  11, 18, 11, 5, 2, 1,
07:  15, 30, 22, 11, 5, 2, 1,
08:  22, 53, 42, 24, 11, 5, 2, 1,
09:  30, 85, 78, 46, 24, 11, 5, 2, 1,
10:  42, 139, 138, 90, 48, 24, 11, 5, 2, 1,
11:  56, 215, 239, 164, 94, 48, 24, 11, 5, 2, 1,
12:  77, 336, 405, 298, 176, 96, 48, 24, 11, 5, 2, 1,
13:  101, 504, 669, 520, 324, 180, 96, 48, 24, 11, 5, 2, 1,
14:  135, 760, 1088, 899, 580, 336, 182, 96, 48, 24, 11, 5, 2, 1,
15:  176, 1115, 1741, 1512, 1020, 606, 340, 182, 96, 48, 24, 11, 5, 2, 1,
...

Alois P. Heinz, Rows n = 1..141, flattened

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      min(d, k)*d, d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=1..n), n=1..15);  # Alois P. Heinz, Mar 15 2014

(* load EulerTransform from 'seqtranslib.m' under OEIS-Transforms *) Table[EulerTransform[Table[Min[c, r], {r, 20}]] - EulerTransform[Table[Min[c-1, r], {r, 20}]], {c, 20}] // Transpose
(* second program: *)
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[Min[d, k]*d, {d, Divisors[j]}] *A[n-j, k], {j, 1, n}]/n]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k-1] ]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)

k-th column is EulerTransform[1, 2, 3, .., k, k, k, ..]-EulerTransform[1, 2, 3, .., k-1, k-1, k-1, ..]. - Wouter Meeussen, Aug 29 2004

Definition corrected, Joerg Arndt, Jul 21 2014

A091356 Number of planar partitions of n with exactly 2 rows.

Original entry on oeis.org

1, 2, 5, 9, 18, 30, 53, 85, 139, 215, 336, 504, 760, 1115, 1635, 2351, 3375, 4770, 6725, 9368, 13006, 17885, 24510, 33319, 45139, 60743, 81457, 108610, 144334, 190844, 251542, 330082, 431825, 562710, 731154, 946644, 1222305, 1573155, 2019471

Offset: 2

Christian G. Bower, Jan 02 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Column 2 of A091355.

Cf. A000219, A091357, A091358, A091359, A091360.

b:= proc(n, k) option remember; `if`(n=0, 1, add(add(min(d, k)
     *d, d=numtheory[divisors](j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n, 2)-b(n, 1):
seq(a(n), n=2..50);  # Alois P. Heinz, Oct 02 2018

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[Sum[Min[d, k] d, {d, Divisors[j]}] b[n - j, k], {j, 1, n}]/n];
a[n_] :=  b[n, 2] - b[n, 1];
a /@ Range[2, 50] (* Jean-François Alcover, Oct 28 2020, after Alois P. Heinz *)

a(n) = A000990(n) - A000041(n).

cp's OEIS Frontend

A091355 Triangle read by rows: T(n,k) = number of planar partitions of n with k rows.

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