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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=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

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