A237512 Number of solutions to Sum_{k=1..n} k*c(k) = n! , c(k) > 0.

0, 1, 0, 1, 47, 55496, 2080571733, 4441900888487987, 849835826032526606030103, 20540228659655619974131131927286681, 82853643094578125257400348993596774353069331199, 70898139566455107685443806945119782661588205935442233026505921

Offset: 0

Author

Wouter Meeussen, Feb 08 2014

Keywords

• nonn

Comments

a(n) is the number of partitions of n! - n*(n+1)/2 into parts that are at most n. - Alois P. Heinz, Feb 08 2014

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..31
• 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])
• StackExchange, Combinations sum_{k=1..m} k*n_k = m!, Jan 29 2014

Crossrefs

Cf. A236810.

Programs

Mathematica

Table[Coefficient[Series[Product[x^k/(1-x^k),{k,n}],{x,0,n!}],x^(n!) ] ,{n,7}]

Formula

a(n) = [x^(n!)] Product_{k=1..n} x^k/(1-x^k).

a(n) = [x^(n!-n*(n+1)/2)] Product_{k=1..n} 1/(1-x^k). - Alois P. Heinz, Feb 08 2014

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