A328362 Triangle read by rows: T(n,k) is the sum of all parts k in all partitions of n into consecutive parts, (1 <= k <= n).

1, 0, 2, 1, 2, 3, 0, 0, 0, 4, 0, 2, 3, 0, 5, 1, 2, 3, 0, 0, 6, 0, 0, 3, 4, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 0, 2, 3, 8, 5, 0, 0, 0, 9, 1, 2, 3, 4, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 5, 6, 0, 0, 0, 0, 11, 0, 0, 3, 4, 5, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 6, 7, 0, 0, 0, 0, 0, 13, 0, 2, 3, 4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 14

Offset

1,3

Comments

Iff n is a power of 2 (A000079) then row n lists n - 1 zeros together with n.

Iff n is an odd prime (A065091) then row n lists (n - 3)/2 zeros, (n - 1)/2, (n + 1)/2, (n - 3)/2 zeros, n.

Links

Table of n, a(n) for n=1..105.

Formula

T(n,k) = k*A328361(n,k).

Example

Triangle begins:
1;
0, 2;
1, 2, 3;
0, 0, 0, 4;
0, 2, 3, 0, 5;
1, 2, 3, 0, 0, 6;
0, 0, 3, 4, 0, 0, 7;
0, 0, 0, 0, 0, 0, 0, 8;
0, 2, 3, 8, 5, 0, 0, 0, 9;
1, 2, 3, 4, 0, 0, 0, 0, 0, 10;
0, 0, 0, 0, 5, 6, 0, 0, 0,  0, 11;
0, 0, 3, 4, 5, 0, 0, 0, 0,  0,  0, 12;
0, 0, 0, 0, 0, 6, 7, 0, 0,  0,  0,  0, 13;
0, 2, 3, 4, 5, 0, 0, 0, 0,  0,  0,  0,  0, 14;
1, 2, 3, 8,10, 6, 7, 8, 0,  0,  0,  0,  0,  0, 15;
0, 0, 0, 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0,  0, 16;
...
For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [0, 2, 3, 8, 5, 0, 0, 0, 9].

Crossrefs

Row sums give A245579.

Column 1 gives A010054, n => 1.

Leading diagonal gives A000027.

Cf. A000079, A001227, A065091, A138785, A204217, A237048, A237593, A266531, A285898, A285899, A285900, A285914, A286000, A286001, A299765, A328361.

Sequence in context: A138262 A276990 A127510 * A158810 A129391 A129390

Adjacent sequences: A328359 A328360 A328361 * A328363 A328364 A328365

Keyword

nonn,tabl

Author

Omar E. Pol, Oct 20 2019

Status

approved