A365005 Number of ways to write 2 as a nonnegative linear combination of a strict integer partition of n.

0, 1, 1, 2, 1, 2, 4, 4, 5, 6, 9, 10, 13, 15, 19, 23, 28, 33, 40, 47, 56, 67, 78, 92, 108, 126, 146, 171, 198, 229, 264, 305, 350, 403, 460, 527, 603, 687, 781, 889, 1009, 1144, 1295, 1464, 1653, 1866, 2101, 2364, 2659, 2984, 3347, 3752, 4200, 4696, 5248, 5858

Offset

0,4

Comments

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

Links

Table of n, a(n) for n=0..55.

Example

The a(6) = 4 ways:
  0*5 + 2*1
  0*4 + 1*2
  0*3 + 0*2 + 2*1
  0*3 + 1*2 + 0*1

Mathematica

combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[2, ptn], {ptn, Select[IntegerPartitions[n], UnsameQ@@#&]}]], {n, 0, 30}]

Crossrefs

For 1 instead of 2 we have A096765.

Column k = n - 2 of A116861.

Row n = 2 of A364916.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A364350 counts combination-free strict partitions, complement A364839.

A364913 counts combination-full partitions.

Cf. A137719, A237113, A323092, A365004, A364272, A364907, A364910, A364914, A364915, A365002.

Sequence in context: A108802 A023673 A132965 * A022597 A073252 A134005

Adjacent sequences: A365002 A365003 A365004 * A365006 A365007 A365008

Keyword

nonn

Author

Gus Wiseman, Aug 26 2023

Status

approved