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

0,5

Links

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A029056 A226503 A036847 * A029055 A035397 A033295

Adjacent sequences: A282376 A282377 A282378 * A282380 A282381 A282382

Keyword

nonn

Author

Alois P. Heinz, Feb 13 2017

Status

approved