cp's OEIS Frontend

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

%I A365005 #7 Aug 26 2023 18:17:11
%S A365005 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 A365005 146,171,198,229,264,305,350,403,460,527,603,687,781,889,1009,1144,
%U A365005 1295,1464,1653,1866,2101,2364,2659,2984,3347,3752,4200,4696,5248,5858
%N A365005 Number of ways to write 2 as a nonnegative linear combination of a strict integer partition of n.
%C A365005 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 A365005 The a(6) = 4 ways:
%e A365005   0*5 + 2*1
%e A365005   0*4 + 1*2
%e A365005   0*3 + 0*2 + 2*1
%e A365005   0*3 + 1*2 + 0*1
%t A365005 combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365005 Table[Length[Join@@Table[combs[2,ptn], {ptn,Select[IntegerPartitions[n], UnsameQ@@#&]}]],{n,0,30}]
%Y A365005 For 1 instead of 2 we have A096765.
%Y A365005 Column k = n - 2 of A116861.
%Y A365005 Row n = 2 of A364916.
%Y A365005 A000041 counts integer partitions, strict A000009.
%Y A365005 A008284 counts partitions by length, strict A008289.
%Y A365005 A364350 counts combination-free strict partitions, complement A364839.
%Y A365005 A364913 counts combination-full partitions.
%Y A365005 Cf. A137719, A237113, A323092, A365004, A364272, A364907, A364910, A364914, A364915, A365002.
%K A365005 nonn
%O A365005 0,4
%A A365005 _Gus Wiseman_, Aug 26 2023