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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. 13
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 (list; table; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A258651 A350530 A258850 * A122950 A116489 A166373
KEYWORD
nonn,look,tabl
AUTHOR
Alois P. Heinz, Apr 12 2012
STATUS
approved

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Last modified February 28 20:40 EST 2024. Contains 370400 sequences. (Running on oeis4.)