cp's OEIS Frontend

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.

%I A211856 #25 Aug 19 2019 06:32:29
%S A211856 1,1,1,2,3,4,6,8,10,15,20,25,34,44,56,74,94,117,151,190,236,298,370,
%T A211856 455,567,699,853,1050,1282,1555,1898,2299,2770,3351,4035,4837,5811,
%U A211856 6952,8288,9898,11782,13978,16600,19660,23225,27451,32366,38074,44799,52609
%N 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.
%H A211856 Seiichi Manyama, <a href="/A211856/b211856.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A211856 G.f.: Product_{k>0} (1+x^k)^A038548(k). - _Vaclav Kotesovec_, Aug 19 2019
%F A211856 G.f.: Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))). - _Vaclav Kotesovec_, Aug 19 2019
%e A211856 a(0) = 1: 0 = the empty sum.
%e A211856 a(1) = 1: 1 = 1*1.
%e A211856 a(2) = 1: 2 = 1*2.
%e A211856 a(3) = 2: 3 = 1*1 + 1*2 = 1*3.
%e A211856 a(4) = 3: 4 = 2*2 = 1*1 + 1*3 = 1*4.
%e A211856 a(5) = 4: 5 = 1*1 + 2*2 = 1*2 + 1*3 = 1*1 + 1*4 = 1*5.
%e A211856 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
%e A211856 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.
%p A211856 with(numtheory):
%p A211856 b:= proc(n, i) option remember; local c;
%p A211856       c:= ceil(tau(i)/2);
%p A211856       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)
%p A211856        +add(b(n-i*j, i-1) *binomial(c, j), j=1..min(c, n/i))))
%p A211856     end:
%p A211856 a:= n-> b(n, n):
%p A211856 seq(a(n), n=0..60);
%t A211856 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_ *)
%t A211856 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 *)
%Y A211856 Cf. A000005, A006171, A038548, A066739, A182269, A182270, A211857.
%K A211856 nonn
%O A211856 0,4
%A A211856 _Alois P. Heinz_, Apr 22 2012