A236810 Number of solutions to Sum_{k=1..n} k*c(k) = n! , c(k) >= 0.
0, 1, 2, 7, 169, 91606, 2407275335, 4592460368601183, 855163933625625205568537, 20560615981766266405801870502139241, 82864945825700191674729490954631752385038099201, 70899311833745096407560015806403481692583415598602691709750081
Offset: 0
Keywords
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..31
- P. F. Ayuso, J. M. Grau, A. Oller-Marcen, Von Staudt formula for Sum_{z in Z_n[i]} z^k, arXiv preprint arXiv:1402.0333, 2014, Montsh. Math. 178 (2015) 345-359
- Vaclav Kotesovec, Graph - the asymptotic ratio (Total 90 terms were computed with a program by Doron Zeilberger)
- A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], Dec 2011
- StackExchange, Combinations sum_{k=1..m} k*n_k = m!, Jan 29 2014
Programs
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Mathematica
Table[Coefficient[Series[Product[1/(1- x^k),{k,n}],{x,0,n!}],x^(n!)] ,{n,7}]
Formula
a(n) = [x^(n!)] Product_{k=1..n} 1/(1-x^k).
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
Extensions
a(8)-a(11) from Alois P. Heinz, Feb 08 2014
Comments