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 A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k. 53
 1, 0, 1, 0, 2, 0, 3, 0, 4, 1, 0, 5, 2, 0, 6, 5, 0, 7, 8, 0, 8, 14, 0, 9, 20, 1, 0, 10, 30, 2, 0, 11, 40, 5, 0, 12, 55, 10, 0, 13, 70, 18, 0, 14, 91, 30, 0, 15, 112, 49, 0, 16, 140, 74, 1, 0, 17, 168, 110, 2, 0, 18, 204, 158, 5, 0, 19, 240, 221, 10, 0, 20, 285, 302, 20, 0, 21, 330, 407 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(n,k) is number of partitions of n-k^2 into parts of 2 kinds with at most k of each kind. LINKS Alois P. Heinz, Rows n = 0..600, flattened Findstat, St000183: The side length of the Durfee square of an integer partition Eric Weisstein's World of Mathematics, Durfee Square FORMULA T(n,k) = Sum_{i=0..n-k^2} P*(i,k)*P*(n-k^2-i), where P*(n,k) = P(n+k,k) is the number of partitions of n objects into at most k parts. EXAMPLE Triangle starts: 1; 0, 1; 0, 2; 0, 3; 0, 4, 1; 0, 5, 2; 0, 6, 5; 0, 7, 8; 0, 8, 14; 0, 9, 20, 1; 0, 10, 30, 2; From Gus Wiseman, Apr 12 2019: (Start) Row n = 9 counts the following partitions: (9) (54) (333) (81) (63) (711) (72) (6111) (432) (51111) (441) (411111) (522) (3111111) (531) (21111111) (621) (111111111) (3222) (3321) (4221) (4311) (5211) (22221) (32211) (33111) (42111) (222111) (321111) (2211111) (End) MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))) end: T:= (n, k)-> add(b(m, k)*b(n-k^2-m, k), m=0..n-k^2): seq(seq(T(n, k), k=0..floor(sqrt(n))), n=0..30); # Alois P. Heinz, Apr 09 2012 MATHEMATICA b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; T[n_, k_] := Sum[b[m, k]*b[n-k^2-m, k], {m, 0, n-k^2}]; Table[ T[n, k], {n, 0, 30}, {k, 0, Sqrt[n]}] // Flatten (* Jean-François Alcover, Dec 03 2015, after Alois P. Heinz *) durf[ptn_]:=Length[Select[Range[Length[ptn]], ptn[[#]]>=#&]]; Table[Length[Select[IntegerPartitions[n], durf[#]==k&]], {n, 0, 10}, {k, 0, Sqrt[n]}] (* Gus Wiseman, Apr 12 2019 *) CROSSREFS For a version without zeros see A115994. Row lengths are A003059. Row sums are A000041. Column k = 2 is A006918. Column k = 3 is A117485. Cf. A008284, A115721, A115722, A257990, A325164. Related triangles are A096771, A325188, A325189, A325192, with Heinz-encoded versions A263297, A325169, A065770, A325178. Sequence in context: A073739 A223707 A046767 * A053120 A366601 A336836 Adjacent sequences: A115717 A115718 A115719 * A115721 A115722 A115723 KEYWORD nonn,tabf AUTHOR Franklin T. Adams-Watters, Mar 11 2006 STATUS approved

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Last modified February 21 03:52 EST 2024. Contains 370219 sequences. (Running on oeis4.)