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A258466
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Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.
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19
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1, 1, 3, 8, 25, 82, 307, 1256, 5688, 28044, 149598, 855811, 5217604, 33711592, 229798958, 1646312694, 12355368849, 96861178984, 791258781708, 6720627124140, 59234364096426, 540812222095821, 5106663817156741, 49798678280227488, 500857393908312587
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OFFSET
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0,3
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COMMENTS
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Also number of ways of partitioning a multiset with multiplicities some partition of n into disjoint blocks. Example: a(4) = 25: 1111; 111,2; 1112; 11,22; 1122; 11,2,3; 11,23; 112,3; 113,2; 1123; 1,2,3,4; 1,2,34; 1,23,4; 1,24,3; 1,234; 12,3,4; 12,34; 13,2,4; 13,24; 14,2,3; 14,23; 123,4; 124,3; 134,2; 1234. Formula: a(n) is the sum of Bell numbers of lengths of all integer partitions of n. - Gus Wiseman, Feb 17 2016
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LINKS
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FORMULA
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G.f.: Sum_{k>=0} Bell(k) * x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020
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EXAMPLE
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a(3) = 8: 1a1a1a, 2a1a, 3a, 1a1a1b, 1a1b1a, 1a1b1b, 2a1b, 1a1b1c (in this example the sorts are labeled a, b, c).
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..25);
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MATHEMATICA
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Table[Plus @@ BellB /@ Length /@ IntegerPartitions[n], {n, 0, 24}] (* Gus Wiseman, Feb 17 2016 *)
b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; T[n_, k_] := Sum[b[n, n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; a[n_] := Sum[T[n, k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 01 2016, after Alois P. Heinz *)
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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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