A258466 Number of partitions of n into parts of sorts {1, 2, ... } which are introduced in ascending order.

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

Offset: 0

Author

Alois P. Heinz, May 30 2015

Keywords

• nonn

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

Examples

			a(3) = 8: 1a1a1a, 2a1a, 3a, 1a1a1b, 1a1b1a, 1a1b1b, 2a1b, 1a1b1c (in this example the sorts are labeled a, b, c).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Row sums of A256130.

Cf. A000110, A035310, A262496, A278644, A319731.

Programs

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 *)

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