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A115720 Triangle T(n,k) is the number of partitions of n with Durfee square k. 6
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

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;

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-Fran├žois Alcover, Dec 03 2015, after Alois P. Heinz *)

CROSSREFS

For another version see A115994. Row lengths A003059.

Cf. A115721, A115722, A008284, A006918.

Sequence in context: A073739 A223707 A046767 * A053120 A284976 A008743

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 October 22 14:51 EDT 2018. Contains 316489 sequences. (Running on oeis4.)