A066855 Triangle T(n,k) of numbers of representations of n as a sum of k products of positive integers, k=1..n. 1 is not allowed as a factor, unless it is the only factor.Representations which differ only in the order of terms or factors are considered equivalent.

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 4, 4, 2, 1, 1, 1, 5, 5, 4, 2, 1, 1, 3, 7, 8, 6, 4, 2, 1, 1, 2, 8, 11, 9, 6, 4, 2, 1, 1, 2, 11, 16, 14, 10, 6, 4, 2, 1, 1, 1, 11, 20, 20, 15, 10, 6, 4, 2, 1, 1, 4, 15, 28, 29, 23, 16, 10, 6, 4, 2, 1, 1, 1, 16, 33, 39, 33, 24, 16, 10, 6, 4, 2, 1, 1, 2, 19

Offset

1,7

Comments

Row sums give A066739.

Links

Table of n, a(n) for n=1..93.

Formula

G.f.: Product_{m=1..infinity} (1-y*x^m)^(-A001055(m)). T(n, k) = Sum_{pi} Product_{m=1..n} binomial(p(m)+A001055(m)-1, p(m)), where pi runs through all nonnegative solutions of p(1)+2*p(2)+...+n*p(n)=n, p(1)+p(2)+...+p(n)=k.

Example

[1], [1, 1], [1, 1, 1], [2, 2, 1, 1], [1, 3, 2, 1, 1], ... . For n=5, 5 = 4+1 = 2*2+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, giving the batch [1, 3, 2, 1, 1].

Crossrefs

Cf. A001055, A066739.

Sequence in context: A207194 A349670 A086275 * A175685 A331182 A182591

Adjacent sequences: A066852 A066853 A066854 * A066856 A066857 A066858

Keyword

nonn,tabl

Author

Vladeta Jovovic, Jan 21 2002

Status

approved