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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. 3
1, 1, 3, 9, 35, 167, 943, 6379, 48945, 429651, 4189865, 45307601, 535518109, 6883110373, 95435065935, 1420468921893, 22577620176887, 381695573051099, 6837601709298811, 129375694813679215, 2578070946813526485, 53964818587883937807, 1183805926540690127573 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Oct 29 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A077365, A107895, A256124.

Sequence in context: A034428 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

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Last modified April 19 08:34 EDT 2019. Contains 322241 sequences. (Running on oeis4.)