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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-Fran├žois 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,changed

AUTHOR

Alois P. Heinz, May 30 2015

STATUS

approved

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Last modified January 29 08:09 EST 2020. Contains 331337 sequences. (Running on oeis4.)