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A291552
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Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) is the number of partitions of k (A000041).
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1
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1, 2, 5, 11, 25, 52, 110, 221, 444, 868, 1685, 3212, 6082, 11361, 21071, 38693, 70570, 127670, 229557, 409963, 728069, 1285522, 2258318, 3947115, 6867238, 11893648, 20513199, 35235429, 60292928, 102787903, 174620017, 295644893, 498931699, 839367287, 1407864040, 2354559426, 3926878130
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^p(k), where p(k) = [x^k] Product_{k>=1} 1/(1 - x^k).
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EXAMPLE
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Equivalently (Cayley), a(n) = total number of 2-dimensional partitions of all nonnegative integers <= n.
a(3) = 11 because we have:
0...1...2.11.1...3.21.2.111.11.1
.............1........1.....1..1
...............................1
and 1 + 1 + 3 + 6 = 11.
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MAPLE
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with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
end:
a:= proc(n) option remember; b(n)+`if`(n>0, a(n-1), 0) end:
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MATHEMATICA
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nmax = 36; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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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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