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A328368
Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of all positive integers <= n into k consecutive parts.
2
1, 2, 3, 2, 4, 2, 5, 4, 6, 4, 3, 7, 6, 3, 8, 6, 3, 9, 8, 6, 10, 8, 6, 4, 11, 10, 6, 4, 12, 10, 9, 4, 13, 12, 9, 4, 14, 12, 9, 8, 15, 14, 12, 8, 5, 16, 14, 12, 8, 5, 17, 16, 12, 8, 5, 18, 16, 15, 12, 5, 19, 18, 15, 12, 5, 20, 18, 15, 12, 10, 21, 20, 18, 12, 10, 6, 22, 20, 18, 16, 10, 6, 23, 22, 18, 16, 10, 6
OFFSET
1,2
COMMENTS
Column k lists k times every nonzero multiple of k in nondecreasing order.
Column k lists the partial sums of the k-th column of triangle A285914.
EXAMPLE
Triangle begins:
1;
2;
3, 2;
4, 2;
5, 4;
6, 4, 3;
7, 6, 3;
8, 6, 3;
9, 8, 6;
10, 8, 6, 4;
11, 10, 6, 4;
12, 10, 9, 4;
13, 12, 9, 4;
14, 12, 9, 8;
15, 14, 12, 8, 5;
16, 14, 12, 8, 5;
17, 16, 12, 8, 5;
18, 16, 15, 12, 5;
19, 18, 15, 12, 5;
20, 18, 15, 12, 10;
21, 20, 18, 12, 10, 6;
22, 20, 18, 16, 10, 6;
23, 22, 18, 16, 10, 6;
24, 22, 21, 16, 10, 6;
25, 24, 21, 16, 15, 6;
26, 24, 21, 20, 15, 6;
27, 26, 24, 20, 15, 12;
28, 26, 24, 20, 15, 12, 7;
...
PROG
(PARI) tt(n, k) = k*(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 A285899.
Row n has length A003056(n).
Column 1 gives A000027.
Column k starts with k in the row A000217(k).
Sequence in context: A222817 A344324 A309428 * A052297 A365049 A297162
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Nov 02 2019
STATUS
approved