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

Author

Gus Wiseman, Aug 26 2023

Keywords

• nonn

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).

Examples

			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
		

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.

Programs

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}]