A211856 Number of representations of n as a sum of products of distinct pairs of positive integers, considered to be equivalent when terms or factors are reordered.

1, 1, 1, 2, 3, 4, 6, 8, 10, 15, 20, 25, 34, 44, 56, 74, 94, 117, 151, 190, 236, 298, 370, 455, 567, 699, 853, 1050, 1282, 1555, 1898, 2299, 2770, 3351, 4035, 4837, 5811, 6952, 8288, 9898, 11782, 13978, 16600, 19660, 23225, 27451, 32366, 38074, 44799, 52609

Offset

0,4

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). - Vaclav Kotesovec, Aug 19 2019

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

Example

a(0) = 1: 0 = the empty sum.
a(1) = 1: 1 = 1*1.
a(2) = 1: 2 = 1*2.
a(3) = 2: 3 = 1*1 + 1*2 = 1*3.
a(4) = 3: 4 = 2*2 = 1*1 + 1*3 = 1*4.
a(5) = 4: 5 = 1*1 + 2*2 = 1*2 + 1*3 = 1*1 + 1*4 = 1*5.
a(6) = 6: 6 = 1*1 + 1*5 = 1*1 + 1*2 + 1*3 = 1*2 + 1*4 = 1*2 + 2*2 = 1*6 = 2*3
a(7) = 8: 7 = 1*1 + 1*2 + 1*4 = 1*1 + 1*2 + 2*2 = 1*1 + 1*6 = 1*1 + 2*3 = 1*2 + 1*5 = 1*3 + 1*4 = 1*3 + 2*2 = 1*7.

Maple

with(numtheory):
b:= proc(n, i) option remember; local c;
      c:= ceil(tau(i)/2);
      `if`(n=0, 1, `if`(i<1, 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..60);

Mathematica

b[n_, i_] := b[n, i] = Module[{c}, c = Ceiling[DivisorSigma[0, i]/2]; If[n == 0, 1, If[i < 1, 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, 60}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[Product[Product[(1 + x^(k*j)), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2019 *)

Crossrefs

Cf. A000005, A006171, A038548, A066739, A182269, A182270, A211857.

Sequence in context: A297417 A343502 A238876 * A066816 A247334 A237450

Adjacent sequences: A211853 A211854 A211855 * A211857 A211858 A211859

Keyword

nonn

Author

Alois P. Heinz, Apr 22 2012

Status

approved