A285891 Triangle read by rows: T(n,k) = n*A237048(n,k).

1, 2, 3, 3, 4, 0, 5, 5, 6, 0, 6, 7, 7, 0, 8, 0, 0, 9, 9, 9, 10, 0, 0, 10, 11, 11, 0, 0, 12, 0, 12, 0, 13, 13, 0, 0, 14, 0, 0, 14, 15, 15, 15, 0, 15, 16, 0, 0, 0, 0, 17, 17, 0, 0, 0, 18, 0, 18, 18, 0, 19, 19, 0, 0, 0, 20, 0, 0, 0, 20, 21, 21, 21, 0, 0, 21, 22, 0, 0, 22, 0, 0, 23, 23, 0, 0, 0, 0, 24, 0, 24, 0, 0, 0

Offset: 1

Author

Omar E. Pol, May 02 2017

Keywords

• nonn
• tabf

Comments

Conjecture: T(n,k) = n, is also the sum of the parts of the partition of n into k consecutive parts, if such a partition exists, otherwise T(n,k) = 0.

Examples

			Triangle begins:
1;
2;
3,   3;
4,   0;
5,   5;
6,   0,  6;
7,   7,  0;
8,   0,  0;
9,   9,  9;
10,  0,  0, 10;
11, 11,  0,  0;
12,  0, 12,  0;
13, 13,  0,  0;
14,  0,  0, 14;
15, 15, 15,  0, 15;
16,  0,  0,  0,  0;
17, 17,  0,  0,  0;
18,  0, 18, 18,  0;
19, 19,  0,  0,  0;
20,  0,  0,  0, 20;
21, 21, 21,  0,  0, 21;
22,  0,  0, 22,  0,  0;
23, 23,  0,  0,  0,  0;
24,  0, 24,  0,  0,  0;
25, 25,  0,  0, 25,  0;
26,  0,  0, 26,  0,  0;
27, 27, 27,  0,  0, 27;
28,  0,  0,  0,  0,  0, 28;
...
		

Crossrefs

Row sums give A245579.

Row n has length A003056(n).

Column k starts in row A000217(k).

The number of positive terms in row n is A001227(n), the number of partitions of n into consecutive parts.

Cf. A196020, A211343, A235791, A236104, A237048, A237591, A237593, A245579, A285900, A285914, A286013.

Programs

PARI

t(n, k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0); \\ A237048
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(n*t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 04 2019