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A282249
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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.
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2
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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, 25, 25, 30, 34, 33, 42, 45, 41, 55, 62, 58, 66, 79, 76, 94, 95, 97, 115, 131, 120, 148, 153, 159, 175, 203, 189, 226, 232, 243, 268, 299, 271, 340, 349, 363, 389
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OFFSET
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0,7
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COMMENTS
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Or number of partitions of n where part i has multiplicity < i and all multiplicities are distinct and different from all parts.
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LINKS
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EXAMPLE
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a(0) = 1: the empty sum.
a(6) = 2: 1*6 = 2*3.
a(8) = 2: 1*8 = 2*4.
a(10) = 3: 1*10 = 2*5 = 1*4+2*3.
a(11) = 3: 1*11 = 1*5+2*3 = 2*4+1*3.
a(12) = 4: 1*12 = 2*6 = 1*6+2*3 = 3*4.
a(13) = 4: 1*13 = 1*7+2*3 = 2*5+1*3 = 1*5+2*4.
a(14) = 6: 1*14 = 1*8+2*3 = 2*7 = 1*6+2*4 = 2*5+1*4 = 3*4+1*2.
a(15) = 5: 1*15 = 1*9+2*3 = 1*7+2*4 = 2*6+1*3 = 3*5.
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.
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MAPLE
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h:= proc(n) option remember;
(((2*n+3)*n-2)*n-`if`(n::odd, 3, 0))/12
end:
g:= (n, i, s)-> `if`(n=0, 1, `if`(n>h(i), 0,
b(n, i, select(x-> x<=i, s)))):
b:= proc(n, i, s) option remember; g(n, i-1, s)+
`if`(i in s, 0, add(`if`(j in s, 0, g(n-i*j,
min(n-i*j, i-1), s union {j})), j=1..min(i-1, n/i)))
end:
a:= n-> g(n$2, {}):
seq(a(n), n=0..100);
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MATHEMATICA
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h[n_] := h[n] = (((2*n + 3)*n - 2)*n - If[OddQ[n], 3, 0])/12;
g[n_, i_, s_] := If[n==0, 1, If[n>h[i], 0, b[n, i, Select[s, # <= i&]]]];
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]}]];
a[n_] := g[n, n, {}];
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CROSSREFS
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Cf. A000009, A066739, A098859, A182269, A182270, A211856, A211857, A212214, A212215, A212216, A212217, A212218, A212219, A276429, A282379.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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