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%I #28 Aug 10 2021 16:14:55
%S 1,1,2,4,11,37,168,926,6181,47651,418546,4106264,44537519,528408261,
%T 6807428748,94588717554,1409927483625,22437711255279,379674820846534,
%U 6806486383431340,128862216628864163,2569080120361323721,53797824318887051264,1180533584545138213222
%N For each partition of n, calculate (dM2/dM3) where dM2 = A036039(p) and dM3 = A036040(p); then sum over all partitions of n.
%C 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.
%H Vaclav Kotesovec, <a href="/A107107/b107107.txt">Table of n, a(n) for n = 0..448</a>
%F For partition [<c_i^k_i>], the contribution to the sum is product_i (c_i - 1)!^k_i.
%F G.f.: 1/Product_{m>0} (1-(m-1)!*x^m). - _Vladeta Jovovic_, Jul 10 2007
%F 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
%F 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
%e 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)
%e equals (120,24,6,4,6,2,1,2,1,1,1) so A107107(6) = 168.
%p b:= proc(n, i) option remember;
%p `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
%p `if`(i>n, 0, b(n-i, i)*(i-1)!)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, May 11 2016
%t nmax=20; CoefficientList[Series[Product[1/(1-(k-1)!*x^k),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Mar 15 2015 *)
%o (Maxima)
%o 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)!;
%o makelist(S(n,1),n,1,17); /* _Vladimir Kruchinin_, Sep 07 2014 */
%Y Cf. A000142, A036039, A000110, A036040, A107106, A102189.
%Y Cf. A077365.
%K easy,nonn
%O 0,3
%A _Alford Arnold_, May 12 2005
%E Edited, corrected and extended by _Franklin T. Adams-Watters_, Nov 03 2005
%E More terms from _Vladeta Jovovic_, Jul 10 2007
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