A278644 Number of partitions of n into parts of sorts {1, 2, ... }.

1, 1, 4, 17, 95, 649, 5423, 53345, 604570, 7744990, 110596370, 1740967790, 29943077149, 558541778035, 11229820022013, 242071441524480, 5568954194762675, 136181762611151941, 3527284819779421843, 96465042641948254298, 2777679881076121497601

Offset: 0

Author

Alois P. Heinz, Nov 24 2016

Keywords

• nonn

Comments

Parts are unordered, sorts are ordered, all sorts up to the highest have to be present.

a(n) mod 2 = A040051(n).

Examples

			a(3) = 17: 1a1a1a, 2a1a, 1a, 1a1a1b, 1a1b1a, 1b1a1a, 1b1b1a, 1b1a1b, 1a1b1b, 2a1b, 2b1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a (in this example the sorts are labeled a, b, c).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..424

Crossrefs

Row sums of A255970.

Cf. A000670, A008284, A040051, A258466.

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:
a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..25);

Mathematica

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]]]]; a[n_] := Sum[Sum[b[n, n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 06 2017, translated from Maple *)

Formula

a(n) = Sum_{k=0..n} A255970(n,k).

a(n) = Sum_{k=0..n} A008284(n,k) * A000670(k). - Ludovic Schwob, Sep 25 2023

a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Sep 26 2023