A182114 Number of partitions of n with largest inscribed rectangle having area <= k; triangle T(n,k), 0<=n, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 2, 5, 0, 0, 0, 1, 5, 7, 0, 0, 0, 0, 5, 7, 11, 0, 0, 0, 0, 3, 7, 13, 15, 0, 0, 0, 0, 1, 5, 16, 18, 22, 0, 0, 0, 0, 0, 3, 17, 21, 27, 30, 0, 0, 0, 0, 0, 1, 16, 22, 34, 38, 42, 0, 0, 0, 0, 0, 0, 13, 21, 39, 48, 54, 56

Offset

0,6

Comments

T(n,k) = A000041(k) for n<k is omitted from the triangle.

Sum_{n>=0} T(n,k) = A115725(k).

Links

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

Formula

T(n,k) = Sum_{i=1..k} A115723(n,i) for n>0, T(0,0) = 1.

Example

T(5,4) = 5 because there are 5 partitions of 5 with largest inscribed rectangle having area <= 4: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4].
T(9,5) = 3: [1,1,1,2,4], [1,1,1,1,5], [1,1,2,5].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 3;
  0, 0, 0, 2, 5;
  0, 0, 0, 1, 5, 7;
  0, 0, 0, 0, 5, 7, 11;
  0, 0, 0, 0, 3, 7, 13, 15;
  0, 0, 0, 0, 1, 5, 16, 18, 22;
  0, 0, 0, 0, 0, 3, 17, 21, 27, 30;
  0, 0, 0, 0, 0, 1, 16, 22, 34, 38, 42;
  ...

Maple

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i=1, `if`(t+n<=k, 1, 0), `if`(i<1, 0, b(n, i-1, t, k)+
       add(`if`(t+j<=k/i, b(n-i*j, i-1, t+j, k), 0), j=1..n/i))))
    end:
T:= (n, k)-> b(n, n, 0, k):
seq(seq(T(n, k), k=0..n), n=0..15);

Mathematica

b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i == 1, If[t + n <= k, 1, 0], If[i < 1, 0, b[n, i - 1, t, k] + Sum[If[t + j <= k/i, b[n - i*j, i - 1, t + j, k], 0], {j, 1, n/i}]]]] ; T[n_, k_] := b[n, n, 0, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 15}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Crossrefs

Diagonal gives: A000041.

T(n,n-1) = A144300(n) = A000041(n) - A000005(n).

T(n+d,n) for d=2-10 give: A218623, A218624, A218625, A218626, A218627, A218628, A218629, A218630, A218631.

Cf. A115723, A115725.

Sequence in context: A258651 A350530 A258850 * A122950 A116489 A166373

Adjacent sequences: A182111 A182112 A182113 * A182115 A182116 A182117

Keyword

nonn,look,tabl

Author

Alois P. Heinz, Apr 12 2012

Status

approved