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

Author

Wouter Meeussen, Feb 08 2014

Keywords

• nonn

Comments

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

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

Crossrefs

Cf. A008290, A008637, A237512, A258668, A258669, A258670, A258671.

Programs

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