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

1,2

Comments

Row sums give A000219.

Columns 1-5 are respectively A000041, A091356, A091357, A091358, and A091359.

Columns converge to A091360.

Links

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

Formula

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

Example

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,
...

Maple

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

Mathematica

(* 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 *)

Crossrefs

Sequence in context: A171730 A131243 A038497 * A131245 A179927 A104446

Adjacent sequences: A091352 A091353 A091354 * A091356 A091357 A091358

Keyword

nonn,tabl

Author

Christian G. Bower, Jan 02 2004

Extensions

Definition corrected, Joerg Arndt, Jul 21 2014

Status

approved