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A144267
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Partition number array, called M32(-4), related to A011801(n,m)= |S2(-4;n,m)| ( generalized Stirling triangle).
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5
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1, 4, 1, 36, 12, 1, 504, 144, 48, 24, 1, 9576, 2520, 1440, 360, 240, 40, 1, 229824, 57456, 30240, 12960, 7560, 8640, 960, 720, 720, 60, 1, 6664896, 1608768, 804384, 635040, 201096, 211680, 90720, 60480, 17640, 30240, 6720, 1260, 1680, 84, 1, 226606464, 53319168
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OFFSET
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1,2
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COMMENTS
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Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k)=:M32(-4;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+3)-ary trees if the outdegree is r >= 0.
If M32(-4;n,k) is summed over those k with fixed number of parts m one obtains triangle A011801(n,m)= |S2(-4;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.
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LINKS
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FORMULA
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a(n,k) = (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S2(-4,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-4,j,1)|^e(n,k,j),j=1..n), with |S2(-4,n,1)|= A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if n=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).
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EXAMPLE
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a(4,3)=48. The relevant partition of 4 is (2^2). The 48 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are quaternary because r=1 vertices are quaternary (4-ary) and for the leaves (r=0) the arity does not matter. Each of the three differently labeled forests comes therefore in 4^2=16 versions due to the two quaternary root vertices.
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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