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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. 0

%I

%S 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,

%T 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,

%U 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

%N 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.

%C Row sums give A066739.

%F 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.

%e [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].

%Y Cf. A001055, A066739.

%K nonn,tabl

%O 1,7

%A _Vladeta Jovovic_, Jan 21 2002

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Last modified June 23 04:15 EDT 2021. Contains 345395 sequences. (Running on oeis4.)