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
Examples
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, ...
Links
- Alois P. Heinz, Rows n = 1..141, flattened
Programs
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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 *)
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
Extensions
Definition corrected, Joerg Arndt, Jul 21 2014
Comments