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

1,2

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Eric Weisstein's World of Mathematics, Durfee Square.

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A008459; row sums A051924.

Sequence in context: A208760 A340156 A340242 * A063858 A209831 A284367

Adjacent sequences: A116644 A116645 A116646 * A116648 A116649 A116650

Keyword

easy,nonn,tabl

Author

Franklin T. Adams-Watters, Feb 20 2006

Status

approved