A365005 - OEIS

%I #7 Aug 26 2023 18:17:11

%S 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,

%T 146,171,198,229,264,305,350,403,460,527,603,687,781,889,1009,1144,

%U 1295,1464,1653,1866,2101,2364,2659,2984,3347,3752,4200,4696,5248,5858

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

%C 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).

%e The a(6) = 4 ways:

%e   0*5 + 2*1

%e   0*4 + 1*2

%e   0*3 + 0*2 + 2*1

%e   0*3 + 1*2 + 0*1

%t combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];

%t Table[Length[Join@@Table[combs[2,ptn], {ptn,Select[IntegerPartitions[n], UnsameQ@@#&]}]],{n,0,30}]

%Y For 1 instead of 2 we have A096765.

%Y Column k = n - 2 of A116861.

%Y Row n = 2 of A364916.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A364350 counts combination-free strict partitions, complement A364839.

%Y A364913 counts combination-full partitions.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Aug 26 2023