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A294617
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Number of ways to choose a set partition of a strict integer partition of n.
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19
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1, 1, 1, 3, 3, 5, 10, 12, 17, 24, 44, 51, 76, 98, 138, 217, 272, 366, 493, 654, 848, 1284, 1560, 2115, 2718, 3610, 4550, 6024, 8230, 10296, 13354, 17144, 21926, 27903, 35556, 44644, 59959, 73456, 94109, 117735, 150078, 185800, 235719, 290818, 365334, 467923
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f.: Sum_{k>=0} Bell(k) * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020
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EXAMPLE
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The a(6) = 10 set partitions are: {{6}}, {{1},{5}}, {{5,1}}, {{2},{4}}, {{4,2}}, {{1},{2},{3}}, {{1},{3,2}}, {{2,1},{3}}, {{3,1},{2}}, {{3,2,1}}.
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, combinat[bell](t), b(n, i-1, t)+
`if`(i>n, 0, b(n-i, min(n-i, i-1), t+1))))
end:
a:= n-> b(n$2, 0):
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MATHEMATICA
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Table[Total[BellB[Length[#]]&/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n, 25}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n > i (i + 1)/2, 0, If[n == 0, BellB[t], b[n, i - 1, t] + If[i > n, 0, b[n - i, Min[n - i, i - 1], t + 1]]]];
a[n_] := b[n, n, 0];
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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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