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

Author

Thomas Wieder, May 26 2005

Keywords

• nonn

Examples

			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.
		

Links

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

Crossrefs

Cf. A077365, A107895, A256124.

Programs

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 *)

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

Extensions

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