A116647 Triangle of number of partitions that fit in an n X n box (but not in an (n-1) X (n-1) box) with Durfee square k.

1, 3, 1, 5, 8, 1, 7, 27, 15, 1, 9, 64, 84, 24, 1, 11, 125, 300, 200, 35, 1, 13, 216, 825, 1000, 405, 48, 1, 15, 343, 1911, 3675, 2695, 735, 63, 1, 17, 512, 3920, 10976, 12740, 6272, 1232, 80, 1, 19, 729, 7344, 28224, 47628, 37044, 13104, 1944, 99, 1, 21, 1000, 12825

Offset: 1

Author

Franklin T. Adams-Watters, Feb 20 2006

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins
   1;
   3,   1;
   5,   8,   1;
   7,  27,  15,   1;
   9,  64,  84,  24,   1;
  11, 125, 300, 200,  35,   1;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Eric Weisstein's World of Mathematics, Durfee Square.

Crossrefs

Cf. A008459; row sums A051924.

Programs

Mathematica

Table[Binomial[n, k]^2 - Binomial[n - 1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Nov 20 2017 *)

PARI

for(n=1,10, for(k=0,n, print1(binomial(n,k)^2 - binomial(n-1,k)^2, ", "))) \\ G. C. Greubel, Nov 20 2017

Formula

T(n,k) = binomial(n,k)^2 - binomial(n-1,k)^2.