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

Author

Vladeta Jovovic, Jan 21 2002

Keywords

• nonn
• tabl

Comments

Row sums give A066739.

Examples

			[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.

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.