%I #66 Sep 02 2018 02:06:19
%S 0,1,2,7,169,91606,2407275335,4592460368601183,
%T 855163933625625205568537,20560615981766266405801870502139241,
%U 82864945825700191674729490954631752385038099201,70899311833745096407560015806403481692583415598602691709750081
%N Number of solutions to Sum_{k=1..n} k*c(k) = n! , c(k) >= 0.
%C a(n) is the number of partitions of n! into parts that are at most n. a(3) = 7: [1,1,1,1,1,1], [2,1,1,1,1], [2,2,1,1], [2,2,2], [3,1,1,1], [3,2,1], [3,3]. - _Alois P. Heinz_, Feb 08 2014
%H Alois P. Heinz, <a href="/A236810/b236810.txt">Table of n, a(n) for n = 0..31</a>
%H P. F. Ayuso, J. M. Grau, A. Oller-Marcen, <a href="http://arxiv.org/abs/1402.0333">Von Staudt formula for Sum_{z in Z_n[i]} z^k</a>, arXiv preprint arXiv:1402.0333, 2014, <a href="http://dx.doi.org/10.1007/s00605-015-0736-5">Montsh. Math. 178 (2015) 345-359</a>
%H Vaclav Kotesovec, <a href="/A236810/a236810_1.jpg">Graph - the asymptotic ratio</a> (Total 90 terms were computed with a program by Doron Zeilberger)
%H A. V. Sills and D. Zeilberger, <a href="http://arxiv.org/abs/1108.4391">Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz)</a>, arXiv:1108.4391 [math.CO], Dec 2011
%H StackExchange, <a href="http://math.stackexchange.com/questions/655541">Combinations sum_{k=1..m} k*n_k = m!</a>, Jan 29 2014
%F a(n) = [x^(n!)] Product_{k=1..n} 1/(1-x^k).
%F a(n) ~ n * (n!)^(n-3) ~ n^(n^2-5*n/2-1/2) * (2*Pi)^((n-3)/2) / exp(n*(n-3)-1/12). - _Vaclav Kotesovec_, Jun 05 2015
%e for n=3, the 7 solutions are: 3! = 6,0,0 ; 4,1,0 ; 2,2,0 ; 0,3,0 ; 3,0,1 ; 1,1,1 ; 0,0,2.
%t Table[Coefficient[Series[Product[1/(1- x^k),{k,n}],{x,0,n!}],x^(n!)] ,{n,7}]
%Y Cf. A008290, A008637, A237512, A258668, A258669, A258670, A258671.
%K nonn
%O 0,3
%A _Wouter Meeussen_, Feb 08 2014
%E a(8)-a(11) from _Alois P. Heinz_, Feb 08 2014