A328371 Irregular triangle read by rows: T(n,k) is the sum of all parts of all partitions of all positive integers <= n into k consecutive parts.

1, 3, 6, 3, 10, 3, 15, 8, 21, 8, 6, 28, 15, 6, 36, 15, 6, 45, 24, 15, 55, 24, 15, 10, 66, 35, 15, 10, 78, 35, 27, 10, 91, 48, 27, 10, 105, 48, 27, 24, 120, 63, 42, 24, 15, 136, 63, 42, 24, 15, 153, 80, 42, 24, 15, 171, 80, 60, 42, 15, 190, 99, 60, 42, 15, 210, 99, 60, 42, 35, 231, 120, 81, 42, 35, 21

Offset

1,2

Comments

Column k lists the partial sums of the k-th column of triangle A285891.

Links

Table of n, a(n) for n=1..76.

Example

Triangle begins:
    1;
    3;
    6,   3;
   10,   3;
   15,   8;
   21,   8,   6;
   28,  15,   6;
   36,  15,   6;
   45,  24,  15;
   55,  24,  15, 10;
   66,  35,  15, 10;
   78,  35,  27, 10;
   91,  48,  27, 10;
  105,  48,  27, 24,
  120,  63,  42, 24, 15;
  136,  63,  42, 24, 15;
  153,  80,  42, 24, 15;
  171,  80,  60, 42, 15;
  190,  99,  60, 42, 15;
  210,  99,  60, 42, 35;
  231, 120,  81, 42, 35, 21;
  253, 120,  81, 64, 35, 21;
  276, 143,  81, 64, 35, 21;
  300, 143, 105, 64, 35, 21;
  325, 168, 105, 64, 60, 21;
  351, 168, 105, 90, 60, 21;
  378, 195, 132, 90, 60, 48;
  406, 195, 132, 90, 60, 48, 28;
...

Prog

(PARI) tt(n, k) = n*(if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0)); \\ A285891
t(n, k) = sum(j=k*(k+1)/2, n, tt(j, k));
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 04 2019

Crossrefs

Row sums give A285900.

Row n has length A003056(n).

Column 1 gives the nonzero terms of A000217.

Column k starts with A000217(k) in the row A000217(k).

Cf. A196020, A211343, A235791, A236104, A235791, A237048, A237591, A237593, A245579, A262612, A285899, A285914, A285891, A286000, A286001, A286013, A299765, A328362, A328365, A328368.

Sequence in context: A260303 A055373 A263333 * A134440 A129529 A298263

Adjacent sequences: A328368 A328369 A328370 * A328372 A328373 A328374

Keyword

nonn,tabf

Author

Omar E. Pol, Nov 02 2019

Status

approved