A282249 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.

1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 3, 4, 4, 6, 5, 6, 8, 8, 9, 11, 10, 14, 15, 14, 14, 21, 18, 21, 25, 25, 30, 34, 33, 42, 45, 41, 55, 62, 58, 66, 79, 76, 94, 95, 97, 115, 131, 120, 148, 153, 159, 175, 203, 189, 226, 232, 243, 268, 299, 271, 340, 349, 363, 389

Offset

0,7

Comments

Or number of partitions of n where part i has multiplicity < i and all multiplicities are distinct and different from all parts.

Links

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

Example

a(0) = 1: the empty sum.
a(6) = 2: 1*6 = 2*3.
a(8) = 2: 1*8 = 2*4.
a(10) = 3: 1*10 = 2*5 = 1*4+2*3.
a(11) = 3: 1*11 = 1*5+2*3 = 2*4+1*3.
a(12) = 4: 1*12 = 2*6 = 1*6+2*3 = 3*4.
a(13) = 4: 1*13 = 1*7+2*3 = 2*5+1*3 = 1*5+2*4.
a(14) = 6: 1*14 = 1*8+2*3 = 2*7 = 1*6+2*4 = 2*5+1*4 = 3*4+1*2.
a(15) = 5: 1*15 = 1*9+2*3 = 1*7+2*4 = 2*6+1*3 = 3*5.
a(25) = 14: 1*25 = 1*19+2*3 = 1*17+2*4 = 1*15+2*5 = 1*13+2*6 = 1*13+3*4 = 2*11+1*3 = 1*11+2*7 = 2*10+1*5 = 1*10+3*5 = 2*9+1*7 = 1*9+2*8 = 3*7+1*4 = 1*7+3*6.

Maple

h:= proc(n) option remember;
      (((2*n+3)*n-2)*n-`if`(n::odd, 3, 0))/12
    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-1, n/i)))
    end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);

Mathematica

h[n_] := h[n] = (((2*n + 3)*n - 2)*n - If[OddQ[n], 3, 0])/12;
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 - 1, n/i]}]];
a[n_] := g[n, n, {}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 01 2018, after Alois P. Heinz *)

Crossrefs

Cf. A000009, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282379.

Sequence in context: A029197 A029174 A058753 * A330953 A360308 A325702

Adjacent sequences: A282246 A282247 A282248 * A282250 A282251 A282252

Keyword

nonn

Author

Alois P. Heinz, Feb 09 2017

Status

approved