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%I #48 Jan 28 2020 16:09:42
%S 1,1,3,8,25,82,307,1256,5688,28044,149598,855811,5217604,33711592,
%T 229798958,1646312694,12355368849,96861178984,791258781708,
%U 6720627124140,59234364096426,540812222095821,5106663817156741,49798678280227488,500857393908312587
%N Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.
%C 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
%H Alois P. Heinz, <a href="/A258466/b258466.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) = Sum_{k=0..n} A256130(n,k).
%F a(n) ~ Bell(n) = A000110(n). - _Vaclav Kotesovec_, Jun 01 2015
%F G.f.: Sum_{k>=0} Bell(k) * x^k / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Jan 28 2020
%e a(3) = 8: 1a1a1a, 2a1a, 3a, 1a1a1b, 1a1b1a, 1a1b1b, 2a1b, 1a1b1c (in this example the sorts are labeled a, b, c).
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p end:
%p T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
%p a:= n-> add(T(n, k), k=0..n):
%p seq(a(n), n=0..25);
%t Table[Plus @@ BellB /@ Length /@ IntegerPartitions[n], {n, 0, 24}] (* _Gus Wiseman_, Feb 17 2016 *)
%t 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_ *)
%Y Row sums of A256130.
%Y Cf. A000110, A035310, A262496, A278644, A319731.
%K nonn
%O 0,3
%A _Alois P. Heinz_, May 30 2015
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