A330377 Irregular triangle read by rows: T(n,k) is the sum of all parts of all partitions of n with Durfee square of size k, (n >= 1; 1 <= k <= floor(sqrt(n))).

1, 4, 9, 16, 4, 25, 10, 36, 30, 49, 56, 64, 112, 81, 180, 9, 100, 300, 20, 121, 440, 55, 144, 660, 120, 169, 910, 234, 196, 1274, 420, 225, 1680, 735, 256, 2240, 1184, 16, 289, 2856, 1870, 34, 324, 3672, 2844, 90, 361, 4560, 4199, 190, 400, 5700, 6040, 400, 441, 6930, 8547, 714, 484, 8470, 11792, 1298

Offset

1,2

Links

Table of n, a(n) for n=1..62.

Formula

T(n,k) = n*A115994(n,k).

Example

Triangle begins:
    1;
    4;
    9;
   16,    4;
   25,   10;
   36,   30;
   49,   56;
   64,  112;
   81,  180,     9;
  100,  300,    20;
  121,  440,    55;
  144,  660,   120;
  169,  910,   234;
  196, 1274,   420;
  225, 1680,   735;
  256, 2240,  1184,   16;
  289, 2856,  1870,   34;
  324, 3672,  2844,   90;
  361, 4560,  4199,  190;
  400, 5700,  6040,  400;
  441, 6930,  8547,  714;
  484, 8470, 11792, 1298;
...

Crossrefs

Row sums give A066186.

Row lengths give A000196, n >= 1.

Column 1 gives A000290, n >= 1.

Column k starts at row k^2 with k^2.

Cf. A115994.

Sequence in context: A120866 A070443 A279403 * A070643 A070442 A086957

Adjacent sequences: A330374 A330375 A330376 * A330378 A330379 A330380

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 22 2019

Status

approved