A107894 Sum over the products of factorials of parts in all partitions of n where the sum runs over the number of different parts only.

1, 1, 3, 9, 35, 167, 943, 6379, 48945, 429651, 4189865, 45307601, 535518109, 6883110373, 95435065935, 1420468921893, 22577620176887, 381695573051099, 6837601709298811, 129375694813679215, 2578070946813526485, 53964818587883937807, 1183805926540690127573

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Formula

a(n) ~ n! * (1 + 1/n + 3/n^2 + 12/n^3 + 65/n^4 + 443/n^5 + 3626/n^6 + 34811/n^7 + 384479/n^8 + 4806098/n^9 + 67109281/n^10), for coefficients see A256124. - Vaclav Kotesovec, Mar 15 2015

Example

The partitions of 5 are 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+2+2, 1+4, 2+3, 5, the corresponding products of factorials of parts are (when multiple parts are counted once only) 1!, 1!*2!, 1!*3!, 1!*2!, 1!*4!, 2!*3!, 5! and their sum is a(5) = 167.

Maple

b:= proc(n, i) option remember;
      `if`(n=0 or i<2, 1, b(n, i-1) +i!*add(b(n-i*j, i-1), j=1..n/i))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 04 2012

Mathematica

Total[Times@@@(Union/@IntegerPartitions[#]!)]&/@Range[20]  (* Harvey P. Dale, Feb 26 2011 *)
b[n_, i_] := b[n, i] = If[n==0 || i<2, 1, b[n, i-1] + i!*Sum[b[n-i*j, i-1], {j, 1, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)

Crossrefs

Cf. A077365, A107895, A256124.

Sequence in context: A370341 A101880 A222398 * A155858 A000834 A005346

Adjacent sequences: A107891 A107892 A107893 * A107895 A107896 A107897

Keyword

nonn

Author

Thomas Wieder, May 26 2005

Extensions

a(0) inserted and more terms from Alois P. Heinz, Apr 04 2012

Status

approved