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%I #20 May 01 2018 04:09:56
%S 1,0,1,1,1,1,2,1,2,1,3,3,4,4,6,5,6,8,8,9,11,10,14,15,14,14,21,18,21,
%T 25,25,30,34,33,42,45,41,55,62,58,66,79,76,94,95,97,115,131,120,148,
%U 153,159,175,203,189,226,232,243,268,299,271,340,349,363,389
%N Number of representations of n as a sum of products of pairs of positive integers: n = Sum_{k=1..m} i_k*j_k with m >= 0, i_k < j_k, j_k > j_{k+1} and all factors distinct.
%C Or number of partitions of n where part i has multiplicity < i and all multiplicities are distinct and different from all parts.
%H Alois P. Heinz, <a href="/A282249/b282249.txt">Table of n, a(n) for n = 0..1000</a>
%e a(0) = 1: the empty sum.
%e a(6) = 2: 1*6 = 2*3.
%e a(8) = 2: 1*8 = 2*4.
%e a(10) = 3: 1*10 = 2*5 = 1*4+2*3.
%e a(11) = 3: 1*11 = 1*5+2*3 = 2*4+1*3.
%e a(12) = 4: 1*12 = 2*6 = 1*6+2*3 = 3*4.
%e a(13) = 4: 1*13 = 1*7+2*3 = 2*5+1*3 = 1*5+2*4.
%e a(14) = 6: 1*14 = 1*8+2*3 = 2*7 = 1*6+2*4 = 2*5+1*4 = 3*4+1*2.
%e a(15) = 5: 1*15 = 1*9+2*3 = 1*7+2*4 = 2*6+1*3 = 3*5.
%e a(25) = 14: 1*25 = 1*19+2*3 = 1*17+2*4 = 1*15+2*5 = 1*13+2*6 = 1*13+3*4 = 2*11+1*3 = 1*11+2*7 = 2*10+1*5 = 1*10+3*5 = 2*9+1*7 = 1*9+2*8 = 3*7+1*4 = 1*7+3*6.
%p h:= proc(n) option remember;
%p (((2*n+3)*n-2)*n-`if`(n::odd, 3, 0))/12
%p end:
%p g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
%p b(n, i, select(x-> x<=i, s)))):
%p b:= proc(n, i, s) option remember; g(n, i-1, s)+
%p `if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
%p min(n-i*j, i-1), s union {j})), j=1..min(i-1, n/i)))
%p end:
%p a:= n-> g(n$2, {}):
%p seq(a(n), n=0..100);
%t h[n_] := h[n] = (((2*n + 3)*n - 2)*n - If[OddQ[n], 3, 0])/12;
%t g[n_, i_, s_] := If[n==0, 1, If[n>h[i], 0, b[n, i, Select[s, # <= i&]]]];
%t b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + If[MemberQ[s, i], 0, Sum[If[MemberQ[s, j], 0, g[n - i*j, Min[n - i*j, i - 1], s ~Union~ {j}]], {j, 1, Min[i - 1, n/i]}]];
%t a[n_] := g[n, n, {}];
%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, May 01 2018, after _Alois P. Heinz_ *)
%Y Cf. A000009, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282379.
%K nonn
%O 0,7
%A _Alois P. Heinz_, Feb 09 2017
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