A282379 Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k <= j_k, j_k > j_{k+1} and all factors distinct with the exception that i_k = j_k is allowed.

1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 5, 9, 9, 8, 11, 15, 13, 17, 17, 19, 24, 29, 23, 33, 37, 39, 40, 53, 48, 62, 63, 71, 77, 94, 81, 110, 116, 122, 123, 156, 152, 185, 180, 200, 213, 259, 236, 287, 298, 325, 333, 404, 386, 450, 457, 506, 531, 615, 579, 679, 721

Offset: 0

Author

Alois P. Heinz, Feb 13 2017

Keywords

• nonn

Examples

			a(4) = 2: 1*4 = 2*2.
a(5) = 2: 1*5 = 2*2+1*1.
a(6) = 2: 1*6 = 2*3.
a(7) = 3: 1*7 = 2*3+1*1 = 1*3+2*2.
a(8) = 3: 1*8 = 2*4 = 1*4+2*2.
a(9) = 4: 1*9 = 1*5+2*2 = 2*4+1*1 = 3*3.
a(10) = 5: 1*10 = 1*6+2*2 = 2*5 = 1*4+2*3 = 3*3+1*1.
a(11) = 6: 1*11 = 1*7+2*2 = 2*5+1*1 = 1*5+2*3 = 2*4+1*3 = 3*3+1*2.
a(12) = 5: 1*12 = 1*8+2*2 = 2*6 = 1*6+2*3 = 3*4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000009, A000330, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282249.

Programs

Maple

h:= proc(n) option remember;
      n*(n+1)*(2*n+1)/6
    end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
                b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
     `if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
      min(n-i*j, i-1), s union {j})), j=1..min(i, n/i)))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);

Mathematica

h[n_] := h[n] = n(n+1)(2n+1)/6;
g[n_, i_, s_ ] := If[n == 0, 1, If[n > h[i], 0,
     b[n, i, Select[s, # <= i&]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] +
     If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j,
     Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)