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 A258466 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order. 19
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..400 FORMULA a(n) = Sum_{k=0..n} A256130(n,k). a(n) ~ Bell(n) = A000110(n). - Vaclav Kotesovec, Jun 01 2015 G.f.: Sum_{k>=0} Bell(k) * x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020 EXAMPLE a(3) = 8: 1a1a1a, 2a1a, 3a, 1a1a1b, 1a1b1a, 1a1b1b, 2a1b, 1a1b1c (in this example the sorts are labeled a, b, c). MAPLE 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); MATHEMATICA 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 *) CROSSREFS Row sums of A256130. Cf. A000110, A035310, A262496, A278644, A319731. Sequence in context: A197159 A161634 A293385 * A216640 A148794 A143330 Adjacent sequences:  A258463 A258464 A258465 * A258467 A258468 A258469 KEYWORD nonn AUTHOR Alois P. Heinz, May 30 2015 STATUS approved

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Last modified August 1 00:13 EDT 2021. Contains 346377 sequences. (Running on oeis4.)