A211857 Number of representations of n as a sum of products of distinct pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 3, 1, 3, 2, 5, 1, 7, 3, 8, 5, 11, 4, 16, 9, 17, 12, 25, 13, 34, 20, 37, 28, 53, 32, 69, 46, 78, 63, 108, 71, 136, 100, 160, 134, 210, 152, 265, 211, 313, 268, 403, 316, 506, 421, 596, 528, 759, 629, 943, 814, 1111, 1016

Offset

0,11

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Formula

G.f.: Product_{k>0} (1+x^k)^(A038548(k)-1). - Vaclav Kotesovec, Aug 19 2019

G.f.: Product_{i>=1} Product_{j=2..i} (1 + x^(i*j)). - Ilya Gutkovskiy, Sep 23 2019

Example

a(0) = 1: 0 = the empty sum.
a(1) = a(2) = a(3) = 0: no product is < 4.
a(4) = 1: 4 = 2*2.
a(6) = 1: 6 = 2*3.
a(8) = 1: 8 = 2*4.
a(9) = 1: 9 = 3*3.
a(10) = 2: 10 = 2*2 + 2*3 = 2*5.
a(12) = 3: 12 = 2*2 + 2*4 = 2*6 = 3*4.
a(16) = 5: 16 = 2*2 + 2*6 = 2*2 + 3*4 = 2*3 + 2*5 = 2*8 = 4*4.

Maple

with(numtheory):
b:= proc(n, i) option remember; local c;
      c:= ceil(tau(i)/2)-1;
      `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)
       +add(b(n-i*j, i-1) *binomial(c, j), j=1..min(c, n/i))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

Mathematica

b[n_, i_] := b[n, i] = Module[{c}, c = Ceiling[DivisorSigma[0, i]/2]-1; If[n==0, 1, If[i<2, 0, b[n, i-1]+Sum[b[n-i*j, i-1]*Binomial[c, j], {j, 1, Min[c, n/i]}]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

Crossrefs

Cf. A000005, A006171, A038548, A066739, A182269, A182270, A211856.

Sequence in context: A029219 A339373 A212217 * A292803 A348712 A292801

Adjacent sequences: A211854 A211855 A211856 * A211858 A211859 A211860

Keyword

nonn

Author

Alois P. Heinz, Apr 22 2012

Status

approved