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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.
20
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 *)
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 22 2012
STATUS
approved