OFFSET
0,3
COMMENTS
Values for individual partitions (A107106) are factorials when all but one part of the partition has size one or two, but not usually in other cases.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..448
FORMULA
For partition [<c_i^k_i>], the contribution to the sum is product_i (c_i - 1)!^k_i.
G.f.: 1/Product_{m>0} (1-(m-1)!*x^m). - Vladeta Jovovic, Jul 10 2007
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, (k-1)!*S(n-k,k))+(n-1)!, S(n,n)=(n-1)!, S(0,m)=1, S(n,m)=0 for m>n. - Vladimir Kruchinin, Sep 07 2014
a(n) ~ (n-1)! * (1 + 1/n + 3/n^2 + 11/n^3 + 50/n^4 + 278/n^5 + 1861/n^6 + 14815/n^7 + 138477/n^8 + 1497775/n^9 + 18465330/n^10). - Vaclav Kotesovec, Mar 15 2015
EXAMPLE
For n = 6, (120,144,90,40,90,120,15,40,45,15,1) / (1,6,15,10,15,60,15,20,45,15,1)
equals (120,24,6,4,6,2,1,2,1,1,1) so A107107(6) = 168.
MAPLE
b:= proc(n, i) option remember;
`if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
`if`(i>n, 0, b(n-i, i)*(i-1)!)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, May 11 2016
MATHEMATICA
nmax=20; CoefficientList[Series[Product[1/(1-(k-1)!*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 15 2015 *)
PROG
(Maxima)
S(n, m):=if n=0 then 1 else if n<m then 0 else if n=m then (n-1)! else sum((k-1)!*S(n-k, k), k, m, n/2)+(n-1)!;
makelist(S(n, 1), n, 1, 17); /* Vladimir Kruchinin, Sep 07 2014 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Alford Arnold, May 12 2005
EXTENSIONS
Edited, corrected and extended by Franklin T. Adams-Watters, Nov 03 2005
More terms from Vladeta Jovovic, Jul 10 2007
STATUS
approved