A182269 Number of representations of n as a sum of products of pairs of positive integers, considered to be equivalent when terms or factors are reordered.

1, 1, 2, 3, 6, 8, 14, 19, 31, 43, 65, 88, 132, 177, 253, 340, 478, 633, 874, 1150, 1562, 2045, 2736, 3553, 4713, 6082, 7969, 10234, 13301, 16973, 21889, 27789, 35577, 44961, 57179, 71906, 90950, 113874, 143204, 178592, 223505, 277599, 345822, 427934, 530797

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, Transforms

Formula

Euler transform of A038548.

G.f.: Product_{k>0} 1/(1-x^k)^A038548(k).

G.f.: Product_{k>=1} (Product_{j=1..k} 1/(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) = 2: 2 = 1*1 + 1*1 = 1*2.
a(3) = 3: 3 = 1*1 + 1*1 + 1*1 = 1*1 + 1*2 = 1*3.
a(4) = 6: 4 = 1*1 + 1*1 + 1*1 + 1*1 = 1*1 + 1*1 + 1*2 = 1*1 + 1*3 = 1*2 + 1*2 = 2*2 = 1*4.

Maple

with(numtheory):
a:= proc(n) option remember;  `if`(n=0, 1, add(add(
       d*ceil(tau(d)/2), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*Ceiling[DivisorSigma[0, d]/2], {d, Divisors[j]}]*a[n-j], {j, 1, 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/(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, A182270, A211856, A211857.

Sequence in context: A089426 A167934 A327690 * A321360 A321566 A066739

Adjacent sequences: A182266 A182267 A182268 * A182270 A182271 A182272

Keyword

nonn

Author

Alois P. Heinz, Apr 22 2012

Status

approved